Question 1

Compute the result of dividing 3 by one third, that is, evaluate 3 divided by 1/3.

Accepted Answer

Introduction / Context: This problem tests understanding of division involving fractions, particularly the idea that dividing by a fraction is equivalent to multiplying by its reciprocal. Many students remember rules mechanically but still make mistakes when applying them to simple numbers. The calculation 3 divided by 1/3 is a classic example used to reinforce this concept in arithmetic and quantitative aptitude tests. Given Data / Assumptions: - The dividend is 3, a whole number. - The divisor is 1/3, a proper fraction less than 1. - We are asked to compute 3 divided by 1/3 exactly. - All operations are in the usual real number system with standard fraction rules. Concept / Approach: The key rule is that dividing by a nonzero fraction a / b is the same as multiplying by its reciprocal b / a. In symbols, x divided by (a / b) equals x * (b / a), provided a is not zero. Applying this to the present problem, we replace division by 1/3 with multiplication by 3/1. This leads to a straightforward integer multiplication. Interpreting the operation in words as How many one third portions are there in 3 also gives an intuitive check on the result. Step-by-Step Solution: Step 1: Write the expression in fractional form as 3 divided by 1/3. Step 2: Use the rule for dividing by a fraction: 3 divided by 1/3 equals 3 multiplied by the reciprocal of 1/3. Step 3: The reciprocal of 1/3 is 3/1. Step 4: Multiply: 3 * 3/1 = 9/1. Step 5: Simplify 9/1 to 9. Step 6: Therefore the value of 3 divided by 1/3 is 9. Verification / Alternative check: Think of 3 as 3 whole units of something, for example 3 metres of rope. Each piece of size 1/3 metre is one third of a metre. The question 3 divided by 1/3 asks how many pieces of length 1/3 metre can be cut from 3 metres. Each metre can be divided into 3 such pieces, so from 3 metres we get 3 * 3 = 9 equal pieces. This real world interpretation matches the calculated result of 9 and confirms that our arithmetic is correct. Why Other Options Are Wrong: Option 1: This might come from confusing division and multiplication, but 3 divided by 1/3 is much larger than 3, not smaller. Option 0: Division by a positive fraction greater than zero cannot produce zero from a positive whole number, so this is impossible here. Option None of these: Since one of the options matches the correct value 9, this option is not correct. Option 3/9: This fraction simplifies to 1/3, which is actually the divisor, not the result, indicating a misunderstanding of the operation. Common Pitfalls: A common error is to multiply 3 by 1/3 instead of dividing, which gives 1. Another mistake is to invert the wrong fraction or to think that dividing by a fraction always makes the result smaller. In fact, dividing by a fraction less than 1 makes the result larger. Keeping the rule multiply by the reciprocal and treating fractions carefully helps avoid these missteps. Final Answer: The value of 3 divided by 1/3 is 9.

Question 2

If 200 cats kill 200 mice in 200 days at a constant rate, then, under the same conditions, how many days will 8 cats take to kill 8 mice?

Accepted Answer

Introduction / Context: This is a classic work and time style puzzle framed in terms of cats and mice. It tests proportional reasoning and the ability to interpret rates correctly. The data indicate how many cats are needed to kill a certain number of mice in a given time. The question then changes the numbers of cats and mice but expects us to assume constant individual rates of work. Understanding that each cat works at the same rate is crucial to answering correctly. Given Data / Assumptions: - Two hundred cats kill two hundred mice in two hundred days. - Each cat works at the same constant rate of killing mice per day. - We are asked how long eight cats will take to kill eight mice under the same conditions. - The rate of work does not change over time, and there are no other influences. Concept / Approach: We interpret the statement 200 cats kill 200 mice in 200 days in terms of unit rates. Since the numbers of cats and mice are equal, we can simplify the situation to find how many mice one cat can kill in a certain period. The principle of direct proportion applies: if every cat kills mice independently at the same speed, then scaling down both cats and mice by the same factor leaves the required time unchanged. This is similar to many work problems where identical workers perform identical tasks. Step-by-Step Solution: Step 1: The statement 200 cats kill 200 mice in 200 days suggests that the total work is killing 200 mice. Step 2: Because 200 cats together kill 200 mice in 200 days, we can see that each cat effectively kills one mouse in 200 days, assuming work is evenly distributed. Step 3: Express the rate of one cat as 1 mouse per 200 days. Step 4: Now consider eight cats. Together, their combined rate is 8 times the rate of one cat, so they kill 8 mice in 200 days as well. Step 5: Equivalently, we can think of the second situation as scaling the first by a factor of 8/200, reducing both cats and mice in the same proportion. When cats and mice are reduced by the same factor, the time required remains the same. Step 6: Therefore, eight cats will also take 200 days to kill eight mice under the same conditions. Verification / Alternative check: Compute the rate explicitly. Each cat kills 1 mouse in 200 days, which is 1/200 mouse per day. Eight cats therefore kill 8 * 1/200 = 8/200 = 1/25 mouse per day. To kill 8 mice at this combined rate, the time needed is total mice divided by rate, which is 8 divided by 1/25 = 8 * 25 = 200 days. This alternative calculation confirms our earlier reasoning and shows that the time required does not change when both cats and mice are scaled equally. Why Other Options Are Wrong: Option 8 days: This would be correct if the total amount of work were scaled differently, but here each cat still needs 200 days on average to kill a single mouse. Option 800 days: This incorrectly assumes that reducing the number of cats slows the process far more than proportional, which conflicts with the constant rate assumption. Option 1 day: This would require an extremely high killing rate that does not match the given data. Option Cannot be determined: The information is sufficient to find the rate and compute the time, so this option is incorrect. Common Pitfalls: Many learners mistakenly assume that if both the number of cats and the number of mice are reduced, the time must also reduce in the same ratio. However, here the key is that the ratio of cats to mice remains 1 to 1, so the time does not change. Another pitfall is forgetting to compute the rate of one cat and instead guessing based on intuition. Writing down the rate explicitly helps avoid such confusion. Final Answer: Eight cats will take 200 days to kill eight mice.

Question 3

A virus population doubles every 3 minutes and reaches a size of K after 1 hour; at what time before the 1 hour mark was the population equal to K divided by 4?

Accepted Answer

Introduction / Context: This question involves exponential growth, a common concept in population dynamics, compound interest, and many other areas. The virus population doubles at regular intervals, specifically every 3 minutes. We are told that after 1 hour the population is K and asked when it was K/4. Problems like this test understanding of exponential sequences and backward reasoning through the growth process. Given Data / Assumptions: - The virus population doubles once every 3 minutes. - The population after 1 hour, that is after 60 minutes, is K. - The growth process is consistent and follows exact doubling with no delays. - We need the time at which the population was one quarter of K, that is K/4. Concept / Approach: Doubling every fixed interval creates a geometric sequence. Going forward in time multiplies the population by 2 each step, while going backward divides by 2. If we know the population at a later time, we can find earlier values by repeatedly halving. To find when the population was K/4, we identify how many doubling steps separate K and K/4. Since K/4 is two halvings before K (because K divided by 2 is K/2 and dividing again by 2 gives K/4), we move back two doubling intervals from 60 minutes. Each interval is 3 minutes, so two intervals is 6 minutes. Step-by-Step Solution: Step 1: Understand that each doubling interval is 3 minutes long. Step 2: At 60 minutes, the population is K. Step 3: One doubling period earlier, at 60 - 3 = 57 minutes, the population must have been half of K, that is K/2. Step 4: Another doubling period earlier, at 57 - 3 = 54 minutes, the population must have been half of K/2, that is K/4. Step 5: Therefore, the population was K/4 at 54 minutes from the start. Step 6: Among the given options, 54 minutes matches this calculation exactly. Verification / Alternative check: We can check by moving forward in time from 54 minutes. At 54 minutes, suppose the population is K/4. After one doubling interval of 3 minutes, at 57 minutes, the population doubles to 2 * (K/4) = K/2. After another 3 minutes, at 60 minutes, it doubles again to 2 * (K/2) = K. This sequence confirms that starting with K/4 at 54 minutes reaches K at 60 minutes, exactly as stated in the problem. No other option produces this correct chain of doublings. Why Other Options Are Wrong: Option 29 minutes: This time is far earlier than needed. From 29 minutes to 60 minutes there are many doubling intervals, so the growth from that time would far exceed K/4 to K in only two steps. Option 25 minutes: Similar to 29 minutes, this time is too early and would involve too many doublings to reach K at 60 minutes. Option 57 minutes: At 57 minutes the population is K/2, not K/4, because it is just one halving step from K rather than two. Option 45 minutes: This is three doubling periods (9 minutes) earlier than 54 minutes, so the population there would be K/32 when we work backward, not K/4. Common Pitfalls: One common mistake is to divide the total time proportionally instead of by doubling intervals. Some learners incorrectly think K/2 must occur at 30 minutes and K/4 at 15 minutes, but that assumes linear growth rather than exponential growth. Another error is miscounting how many halving steps are required to go from K to K/4. Remember that exponential processes are controlled by the number of doubling intervals, not by the raw time fraction. Final Answer: The virus population was K/4 at 54 minutes.

Question 4

Two circles have centers O and O2 with radii 7 cm and 9 cm respectively and the distance between the centers is 20 cm; PQ is a transverse common tangent to the circles that cuts the line segment OO2 at X; find the length of segment O2X in centimetre…

Accepted Answer

Introduction / Context: This geometry question involves two circles, a transverse common tangent, and the line segment joining the centers. It tests understanding of circle geometry, similar triangles, and relationships between radii, center distance, and tangents. The goal is to find the length of a segment on the line joining the centers where the tangent intersects that line. Given Data / Assumptions: - Circle 1 has center O and radius 7 cm. - Circle 2 has center O2 and radius 9 cm. - The distance between the centers O and O2 is 20 cm. - PQ is a transverse common tangent that touches both circles and intersects OO2 at point X. - We must find the length of O2X in centimetres. - The tangent is such that perpendiculars from O and O2 to the tangent meet at right angles at the respective points of contact. Concept / Approach: When a transverse common tangent touches two circles, the line segment joining the centers intersects the tangent, creating two right triangles with the radii as one leg and segments OX and O2X as the other legs. These triangles are similar because they share an acute angle at X and both are right angled at the points of tangency. From similarity, the ratio of distances from X to the centers equals the ratio of the radii. This gives a proportional relationship between OX and O2X that can be combined with the known total distance OO2 to compute each segment. Step-by-Step Solution: Step 1: Let OX be the distance from O to X and O2X be the distance from O2 to X. Step 2: The total distance between the centers is OO2 = OX + O2X = 20 cm. Step 3: Consider the right triangle formed by center O, point X, and the point of tangency on the first circle. Its legs are OX (along OO2) and the radius 7 cm (perpendicular to the tangent). Step 4: Similarly, consider the right triangle formed by center O2, point X, and the point of tangency on the second circle. Its legs are O2X and radius 9 cm. Step 5: The two triangles share the angle at X and both have right angles at the points of tangency, so they are similar. Step 6: By similarity, the ratio of corresponding sides gives OX / O2X = 7 / 9. Step 7: Let OX = 7k and O2X = 9k for some positive constant k. Then OX + O2X = 7k + 9k = 16k = 20. Step 8: Solve for k: k = 20 / 16 = 5 / 4. Step 9: Compute O2X = 9k = 9 * (5 / 4) = 45 / 4 cm. Verification / Alternative check: We can also compute OX = 7k = 7 * (5 / 4) = 35 / 4 cm. Check that OX + O2X equals the total center distance: 35 / 4 + 45 / 4 = 80 / 4 = 20 cm, which matches the given value of OO2. This confirms that the proportional assignment based on similarity was correct and that the resulting segment lengths satisfy the required conditions. Why Other Options Are Wrong: Option 10: If O2X were 10 cm, then OX would be 10 cm as well, suggesting equal radii, which contradicts the given radii 7 cm and 9 cm and the similarity ratio. Option 6: This value ignores the ratio 7 to 9 and does not satisfy OX + O2X = 20 when combined with a corresponding OX consistent with similarity. Option 35/4: This is actually the length of OX, not O2X, so selecting it confuses the two segments. Option 5: This small value for O2X would require OX to be 15 cm, but then OX / O2X would be 3, which does not match the radii ratio 7 / 9. Common Pitfalls: A common mistake is to assume the tangent is symmetric with respect to the midpoint of OO2, leading to equal segments OX and O2X. Another error is to apply the formula for the length of common tangents rather than using triangle similarity for the intersection point X. Carefully identifying the right triangles and using the ratio of radii to relate distances along OO2 is the correct method here. Final Answer: The length of segment O2X is 45/4 cm.

Question 5

The sum of three numbers is 98; the ratio of the first number to the second is 2 : 3 and the ratio of the second number to the third is 5 : 8; find the value of the second number.

Accepted Answer

Introduction / Context: This question is a typical example of using ratios and sums to determine individual values. When multiple ratios between the same set of numbers are given, they can be combined into a single set of proportional parts. Then the total sum is used to scale these parts to actual numbers. Such problems appear frequently in quantitative aptitude and arithmetic reasoning sections. Given Data / Assumptions: - There are three numbers, say A, B, and C. - Their sum is A + B + C = 98. - The ratio of the first to the second, A : B, is 2 : 3. - The ratio of the second to the third, B : C, is 5 : 8. - All numbers are positive real numbers and are assumed to be consistent with the given ratios. Concept / Approach: We have two separate ratios involving the same variable B. To combine them, we express A and C in terms of B and introduce a common scaling factor. Alternatively, we set A, B, and C as multiples of some base quantity using the ratio chain. Once we obtain a complete combined ratio for A : B : C, we compare the sum of these parts with the given total 98 and find the value of one part. From there, it is straightforward to compute the required number, which is the value of B. Step-by-Step Solution: Step 1: Let A : B = 2 : 3. This means A = 2k and B = 3k for some positive constant k. Step 2: Let B : C = 5 : 8. This means B = 5m and C = 8m for some positive constant m. Step 3: Because B appears in both ratios, we equate the expressions for B: 3k = 5m. Step 4: Solve for k and m in terms of a new constant t. Take k = 5t and m = 3t so that 3k = 15t and 5m = 15t, keeping B consistent. Step 5: Substitute back to express each number. Then A = 2k = 2 * 5t = 10t, B = 3k = 15t, and C = 8m = 8 * 3t = 24t. Step 6: Add the three numbers: A + B + C = 10t + 15t + 24t = 49t. Step 7: Use the given sum 98 to write 49t = 98, which gives t = 2. Step 8: Substitute t = 2 to find B: B = 15t = 15 * 2 = 30. Verification / Alternative check: When t = 2, the numbers are A = 10 * 2 = 20, B = 30, and C = 24 * 2 = 48. Their sum is 20 + 30 + 48 = 98, which matches the given total. The ratio A : B is 20 : 30, which simplifies to 2 : 3, and the ratio B : C is 30 : 48, which simplifies to 5 : 8. Both ratios are satisfied. Therefore, B = 30 is consistent with all pieces of given information. Why Other Options Are Wrong: Option 27: If B were 27, the combined ratio and sum conditions would fail; we cannot create A and C that both honour the ratios and add up to 98. Option 23: This value is too small and does not satisfy the derived proportional structure when we try to reconstruct A and C. Option 33: This value is too large in the combined ratio framework and does not yield a correct total sum of 98. Option 35: Similar to 33, this number does not satisfy both the ratio and sum constraints simultaneously. Common Pitfalls: One frequent error is to combine ratios incorrectly, for example by adding them directly instead of using a common multiple approach. Another mistake is to forget that the same B must satisfy both A : B and B : C simultaneously, which requires consistent scaling. Some learners also attempt guesswork with options instead of setting up the ratios algebraically, which can work occasionally but is unreliable for more complex problems. Final Answer: The value of the second number is 30.

Question 6

Let x = square root of 8 minus square root of 7, y = square root of 6 minus square root of 5, and z = square root of 10 minus 3; determine the correct order relation among x, y, and z from the given options.

Accepted Answer

Introduction / Context: This problem asks us to compare three expressions involving square roots and differences. Such questions are designed to test the ability to estimate and compare irrational numbers without necessarily computing them to high precision. By using approximate values or algebraic reasoning, we can determine the correct inequality relation between x, y, and z. Given Data / Assumptions: - x is defined as square root of 8 minus square root of 7. - y is defined as square root of 6 minus square root of 5. - z is defined as square root of 10 minus 3. - Square roots are taken in their principal (positive) values. - All comparisons are made in the real number system. Concept / Approach: One approach is to approximate each square root using known nearby perfect squares and then compute rough decimal values for x, y, and z. Since all three expressions are small positive numbers, even modest accuracy is usually enough to determine their ordering. Another approach involves manipulating the expressions algebraically, but in this case numerical estimation is both faster and sufficiently precise for an aptitude question. Step-by-Step Solution: Step 1: Approximate square roots using known benchmarks. We know that 2.8 squared is 7.84 and 2.9 squared is 8.41, so square root of 8 is a little less than 2.83. Square root of 7 is a bit more than 2.64. Step 2: From these approximations, x = square root of 8 minus square root of 7 is roughly 2.83 - 2.65, giving about 0.18. Step 3: Approximate square root of 6 and square root of 5. Since 2.4 squared is 5.76 and 2.5 squared is 6.25, square root of 6 is around 2.45, and square root of 5 is around 2.24. Step 4: Then y = square root of 6 minus square root of 5 is approximately 2.45 - 2.24, giving about 0.21. Step 5: Approximate square root of 10. Since 3.1 squared is 9.61 and 3.2 squared is 10.24, square root of 10 is about 3.16. Step 6: Compute z = square root of 10 minus 3, which is about 3.16 - 3.00 = 0.16. Step 7: Comparing the approximate values, we have z around 0.16, x around 0.18, and y around 0.21. Step 8: Therefore z is the smallest, y is the largest, and x lies between them, so the relation is z < x < y. Verification / Alternative check: We can refine the approximations slightly. Using more precise estimates, square root of 8 is about 2.828, square root of 7 is about 2.646, so x is about 0.182. Square root of 6 is about 2.449 and square root of 5 is about 2.236, so y is about 0.213. Square root of 10 is about 3.162, so z is about 0.162. The refined values confirm the ordering z < x < y. This consistency between rough and refined estimates strengthens confidence in the inequality chosen. Why Other Options Are Wrong: Option x > y < z: This states that y is smaller than both x and z, which contradicts the calculations showing that y is actually the largest among the three. Option x < y < z: This implies z is the largest, but z has the smallest approximate value, so this relation is incorrect. Option x > y > z: This says x is the largest and z is the smallest, but in reality y is larger than x. Option z < x < y: This matches our ordering and is correct, so the remaining option z < y < x is necessarily wrong. Option z < y < x: This would place y between z and x, but we have already established that x lies between z and y, not the other way around. Common Pitfalls: A common error is to treat square root of n as roughly equal to n divided by 2 or some other inaccurate heuristic, which leads to wrong comparisons. Another pitfall is to compare the differences of the radicands directly, for example assuming that because 8 - 7 is 1 and 6 - 5 is also 1, the differences must be equal, which is not true for square roots. Careful approximation or algebraic reasoning is needed whenever square roots are involved. Final Answer: The correct order relation is z < x < y.

Question 7

Determine the numeric value represented by sixty two tens, that is, find the number equal to 62 groups of ten.

Accepted Answer

Introduction / Context: This question again tests place value and understanding of tens in the base ten number system. Knowing how many ones are in multiple tens and how to express that as a single number is fundamental arithmetic. Sixty two tens means 62 groups of ten, and we must convert that into standard numeric form. Given Data / Assumptions: - Each ten is equal to ten ones. - We have sixty two tens, which is 62 groups of 10. - The goal is to express this quantity as a single number in standard form. Concept / Approach: To convert groups of ten into ones, we multiply the number of tens by 10. The product gives the total number of ones, which is directly the standard value of the quantity. This relies on the simple fact that the tens place in a decimal number represents multiples of ten. Understanding this helps with reading and writing large numbers and with operations such as multiplication and division by powers of ten. Step-by-Step Solution: Step 1: Interpret the phrase sixty two tens as 62 multiplied by 10, since each ten equals 10 ones. Step 2: Compute 62 * 10. This is easy because multiplying by 10 in decimal notation appends one zero to the right of the number. Step 3: 62 * 10 = 620. Step 4: Therefore, sixty two tens equal 620. Verification / Alternative check: We can also think in place value terms. A number with 6 hundreds and 2 tens would be 620, but here the wording sixty two tens refers to 62 tens, which is 620 ones. Expressing 620 as hundreds, tens, and ones gives 6 hundreds, 2 tens, and 0 ones. This is entirely consistent with 62 groups of ten, reinforcing that the correct standard form is 620. Why Other Options Are Wrong: Option 6: This is much smaller than the expected value and corresponds to 6 ones, not 62 tens. Option 62: This represents sixty two ones, which would be 6 tens and 2 ones, not sixty two tens. Option 2: This is only 2 ones and is completely unrelated to the scale of the question. Option 600: This would be sixty tens, not sixty two tens. Although close in magnitude, it does not match the exact phrase sixty two tens. Common Pitfalls: Some learners misinterpret sixty two tens as the number 62 itself, forgetting the multiplication by ten. Another common error is to approximate, choosing 600 because it is a round figure, instead of performing the simple multiplication. Remember that tens, hundreds, and thousands are just labels indicating how many times 10, 100, or 1000 respectively are being counted. Final Answer: The value of sixty two tens is 620.

Question 8

In database and relational joins, if five tables need to be joined in a straight chain so that each table is joined to the next one once, how many join operations are required?

Accepted Answer

Introduction / Context: This question comes from relational database concepts and SQL join operations. When multiple tables need to be combined in a single query, joins connect rows based on related columns. The number of join operations required to connect a given number of tables in a straight chain follows a simple pattern. Understanding this pattern helps in query planning and optimization. Given Data / Assumptions: - There are 5 distinct tables to be joined. - The phrase join straight suggests a simple chain, where each table is joined to exactly one neighbour, forming a linear sequence. - We assume standard join operations in SQL, such as inner join or left join, without considering special cases like cross joins. - We must count the number of join conditions or join operations needed to connect all tables in one query. Concept / Approach: In a straight join chain, each join operation connects one additional table to the already formed result. Starting from one table, each new join brings in exactly one more table. Therefore, the number of joins needed to link N tables in a single chain is N minus 1. This pattern can be seen by simple counting: two tables need one join, three tables need two joins, and so on. Step-by-Step Solution: Step 1: Begin by considering the simplest case. To join 2 tables in a straight chain, we need 1 join operation between them. Step 2: To add a third table, we take the existing result of the first join and join it with the third table. This requires a second join operation. Step 3: In general, each time we add a new table to the chain, we need one more join. Step 4: Therefore, to connect N tables in a straight chain, we need N - 1 join operations. Step 5: For the given case with 5 tables, the required number of joins is 5 - 1 = 4. Verification / Alternative check: Imagine explicitly naming the tables as T1, T2, T3, T4, and T5. A straight chain join might look like T1 join T2, then join T3, then join T4, and then join T5. Counting these operations: T1 joined with T2 is 1 join, joining that result with T3 is a second join, with T4 is a third join, and with T5 is a fourth join. After 4 joins all 5 tables are connected, confirming that 4 joins are sufficient and necessary. Why Other Options Are Wrong: Option 5: This would correspond to N joins for N tables, which is more than needed. One table does not require a join to itself. Option 6: This is even larger and has no basis in the linear chain pattern of joining tables. Option 10: This might come from confusing joins with all possible pairs of tables in a complete graph, but the question specifies a straight chain, not all pairwise connections. Option 3: This would be fewer than N - 1 joins and would leave at least one table disconnected from the chain. Common Pitfalls: Some learners confuse the number of joins required in a chain with the number of possible table pairs, leading to overcounting. Others mistakenly think that each table requires a join, forgetting that the first table serves as the starting point. Remember the general rule: in a straight chain, the number of joins is always one less than the number of tables being connected. Final Answer: To join 5 tables in a straight chain, 4 join operations are required.

Question 9

In an arithmetic reasoning puzzle, there are n boxes and m fruits. If 3 fruits are placed in each box, 3 fruits remain unused. If 4 fruits are placed in each box, then exactly 1 box remains empty. What are the numbers of boxes and fruits?

Accepted Answer

Introduction / Context: This is a classic arithmetic reasoning and linear equations puzzle involving boxes and fruits. We are given two different ways of distributing fruits into boxes and told what happens in each case: in one arrangement some fruits are left over, and in another arrangement some boxes are left empty. The goal is to translate the English description into algebraic equations in terms of the number of boxes and the number of fruits, then solve these equations systematically to find both unknown quantities. Given Data / Assumptions: - Let n be the number of boxes. - Let m be the total number of fruits. - If 3 fruits are placed in each box, then 3 fruits remain extra, so the fruits used in boxes plus the extra equals m. - If 4 fruits are placed in each box, then 1 box is left empty, meaning only n - 1 boxes are actually filled. - Each box that is used in either arrangement contains the same number of fruits specified in that scenario. Concept / Approach: We convert each verbal condition into an algebraic equation. In the first arrangement, the number of fruits used is 3n and there are 3 fruits left over. Therefore the total fruit count m is 3n + 3. In the second arrangement, only n - 1 boxes are used, each containing 4 fruits, so the total number of fruits m is also 4(n - 1). Since both expressions describe the same total m, we can equate them, obtain a simple linear equation in n, and then compute m. Finally, we check which option matches the pair (n, m). Step-by-Step Solution: Step 1: From the first condition, write m = 3n + 3, because 3 fruits per box across n boxes accounts for 3n fruits and there are 3 extra fruits. Step 2: From the second condition, only n - 1 boxes are filled with 4 fruits each, so m = 4(n - 1). Step 3: Since both expressions equal m, set them equal: 3n + 3 = 4(n - 1). Step 4: Expand the right side: 4(n - 1) = 4n - 4, so we have 3n + 3 = 4n - 4. Step 5: Rearrange: 3n + 3 - 4n + 4 = 0 gives -n + 7 = 0, so n = 7 boxes. Step 6: Substitute n = 7 into m = 3n + 3 to get m = 3 * 7 + 3 = 21 + 3 = 24 fruits. Verification / Alternative check: Check the first condition: With 7 boxes and 24 fruits, placing 3 in each box uses 3 * 7 = 21 fruits, leaving 24 - 21 = 3 fruits extra, which matches the statement. Check the second condition: If we place 4 fruits in each box but have only enough fruits to fill 6 boxes, then 4 * 6 = 24 fruits are used and exactly 1 box is left empty. This matches the second condition perfectly. Hence (7, 24) satisfies both constraints. Why Other Options Are Wrong: Option 3 boxes and 9 fruits: With 3 fruits per box, all fruits are used and none are excess, so the first condition fails. The second condition with 4 fruits per box also cannot be satisfied. Option 6 boxes and 19 fruits: 3 fruits per box would use 18 fruits and leave 1 extra, not 3. Thus it does not fit the first condition. Option 5 boxes and 18 fruits: 3 fruits per box uses all 15 fruits, leaving 3 fruits short of 18, not 3 fruits extra. This again does not match the statement. Common Pitfalls: A common error is misreading excess as meaning extra boxes instead of extra fruits, or misinterpreting an excess box as unused fruits. Another mistake is to treat n and m as interchangeable or to forget that the same total number of fruits must satisfy both scenarios. Carefully defining variables and writing equations before solving prevents such confusion. Final Answer: The correct numbers are 7 boxes and 24 fruits.

Question 10

What is the arithmetic mean (average) of the multiset of numbers 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6 and 7, 7, 7, 7, 7, 7, 7?

Accepted Answer

Introduction / Context: This question tests basic statistics, specifically the computation of an arithmetic mean when numbers are repeated. Rather than listing every number individually, we can take advantage of the fact that each value occurs a known number of times. The arithmetic mean is a measure of central tendency and is computed by dividing the total sum of all data values by the total number of data values. Given Data / Assumptions: - The multiset of numbers is: one 1, two 2s, three 3s, four 4s, five 5s, six 6s, and seven 7s. - All numbers are to be included when calculating the mean, with their repetitions counted appropriately. - The arithmetic mean is defined as (sum of all observations) / (number of observations). Concept / Approach: Instead of writing out each number explicitly, we can use the frequencies directly. For every distinct value v that appears f times, it contributes v * f to the sum and f to the count. We sum these contributions for all distinct values and then divide the total sum by the total count. This is efficient and helps avoid arithmetic mistakes when there are many repetitions. Step-by-Step Solution: Step 1: Count how many times each number appears: 1 appears once, 2 appears twice, 3 appears three times, 4 appears four times, 5 appears five times, 6 appears six times, and 7 appears seven times. Step 2: Compute the total number of observations: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. Step 3: Compute the total sum using value × frequency: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, 6 × 6 = 36, 7 × 7 = 49. Step 4: Add these partial sums: 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140. Step 5: Use the mean formula: mean = total sum / total count = 140 / 28. Step 6: Simplify 140 / 28 = 5. Verification / Alternative check: An alternative way is to notice that the pattern is symmetric from 1 to 7, with increasing frequencies, but computation via value times frequency is already simple and direct. Dividing both numerator and denominator by 7 shows that 140 / 28 = (20 × 7) / (4 × 7) = 20 / 4 = 5. Both methods confirm that the arithmetic mean is 5. Why Other Options Are Wrong: Option 6: This is larger than the majority of the data points and does not match the computed average. Option 7: This would be the maximum value, but the presence of many smaller numbers pulls the average down. Option 9: Far exceeds any individual data point and cannot be the mean. Option None of these: Since 5 is one of the options and matches the correct mean, this option is incorrect. Common Pitfalls: Students sometimes forget to count repetitions correctly or mistakenly divide by the number of distinct values instead of the total number of observations. Another common error is arithmetic slips when summing many terms. Using the value × frequency method systematically, and double-checking sums, reduces the risk of such mistakes. Final Answer: The arithmetic mean of the given multiset of numbers is 5.

Question 11

In basic geometry, which statement best compares a line and a point?

Accepted Answer

Introduction / Context: This conceptual question belongs to the foundations of Euclidean geometry. Understanding the basic undefined terms point, line, and plane, and knowing their dimensional properties, is essential for reasoning about shapes, figures, and geometric relationships. The task is to choose the statement that correctly describes and compares a point and a line, using standard geometric definitions. Given Data / Assumptions: - A point is considered an exact position or location in space with no length, width, or thickness. - A line is a straight one-dimensional figure extending without end in both directions, with length but no thickness. - Standard school-level Euclidean geometry terminology applies. - We are selecting the statement that best captures the dimensional difference between a point and a line, without introducing incorrect additional claims. Concept / Approach: The key concept is geometric dimension. A point is zero-dimensional: it only indicates position, not size. A line is one-dimensional: it has length along one direction but no width or height. We examine each option to see whether it correctly states these facts and avoids incorrect claims, such as saying that a point cannot lie on a plane or that a line has planes on it. Step-by-Step Solution: Step 1: Evaluate option A: It says a point has no dimension and a line has one dimension. This aligns exactly with standard geometric definitions (0-dimensional point, 1-dimensional line). Step 2: Evaluate option B: It claims that a point cannot be on any line segment. This is false because any line segment is made up of infinitely many points and any endpoint is itself a point on the segment. Step 3: Evaluate option C: It incorrectly states that a line has several planes located on it. In Euclidean geometry, infinitely many planes can pass through a given line, but lines do not contain planes. Step 4: Evaluate option D: It claims that a point cannot lie on a plane. This is incorrect; a plane is also made up of infinitely many points, so a point can certainly lie on a plane. Step 5: From these evaluations, option A is the only statement that is both accurate and concise in comparing a line and a point. Verification / Alternative check: Referencing standard geometry: a point is described as that which has no part, and a line as breadthless length. This again reinforces that a point is 0-dimensional and a line is 1-dimensional. None of the other options match textbook definitions, confirming that option A is correct. Why Other Options Are Wrong: Option B: Misrepresents the nature of line segments and points by denying that points belong to segments. Option C: Confuses the relationship between lines and planes. Lines lie in planes, but do not contain planes. Option D: Incorrectly excludes a point from lying on a plane, although planes are sets of points. Option None of these: Not correct because option A is fully accurate and sufficient. Common Pitfalls: Students sometimes mix up the hierarchy of geometric objects, thinking lines consist of segments only and forgetting both lines and planes are sets of points. Another confusion is to think that dimension refers to physical thickness, whereas in geometry it refers to independent directions needed to specify a position. Remembering that points (0D), lines (1D), planes (2D), and solids (3D) form a natural progression helps avoid these mistakes. Final Answer: The correct comparison is that a point has no dimension and a line has one dimension.

Question 12

Decide whether the statement Every integer is a rational number is true or false under the standard mathematical definitions of integers and rational numbers.

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Introduction / Context: This question checks basic number theory and the nested structure of number sets. It asks whether every integer belongs to the set of rational numbers. Understanding the formal definition of a rational number and how integers fit into that definition is essential in algebra and higher mathematics, because many properties and theorems depend on these set relationships. Given Data / Assumptions: - An integer is any whole number, positive, negative, or zero, such as ..., -3, -2, -1, 0, 1, 2, 3, .... - A rational number is any number that can be written in the form p / q where p and q are integers and q is not zero. - Standard school-level definitions are used without any non-standard conventions. Concept / Approach: To test whether every integer is a rational number, we need to see whether an arbitrary integer can always be represented in the form p / q with integer p and q and q not equal to zero. If this is possible for all integers, then the statement is true. If even one integer cannot be written in that fractional form, the statement would be false. We therefore try to represent a general integer n in rational form. Step-by-Step Solution: Step 1: Let n be any integer. It may be positive, negative, or zero. Step 2: Consider the fraction n / 1. Here the numerator n is an integer, and the denominator 1 is also an integer. Step 3: The denominator 1 is not zero, so n / 1 satisfies the requirement for being a rational number. Step 4: The fraction n / 1 is equal to n itself, so every integer n can be written as a rational number without changing its value. Step 5: Since n was arbitrary, this reasoning holds for all integers, including negative integers and zero. Verification / Alternative check: Try specific examples: 5 can be written as 5 / 1; -3 can be written as -3 / 1; 0 can be written as 0 / 1. In each case, numerator and denominator are integers and the denominator is not zero. Therefore, 5, -3, and 0 are all rational numbers. This confirms that the set of integers is fully contained within the set of rational numbers. Why Other Options Are Wrong: Option FALSE: This would require at least one integer to fail the rational number definition, which does not happen. Option Cannot be determined: The definitions are precise, so there is no ambiguity; the statement’s truth can be determined exactly. Option True only for positive integers: Negative integers and zero also fit the form n / 1, so limiting to positive integers is unnecessary and incorrect. Option True only for non-zero integers: Zero is 0 / 1 and is rational, so excluding zero is not justified. Common Pitfalls: Some learners incorrectly think rational means fraction that is not a whole number, and therefore wrongly exclude integers. Others confuse rational with real or with decimal numbers that terminate or repeat. Remember that rational specifically means expressible as a ratio of two integers with a non-zero denominator. Under this definition, every integer is automatically rational. Final Answer: The statement Every integer is a rational number is TRUE.

Question 13

In an arithmetic reasoning puzzle, two examination rooms P and Q contain some students. If 10 students are sent from room P to room Q, the numbers in both rooms become equal. If instead 20 students are sent from room Q to room P, then the number of …

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Introduction / Context: This problem is a typical linear equation word problem involving two groups of students and transfers between them. We are given two different hypothetical movements of students between the rooms and told how the numbers change in each scenario. The goal is to set up algebraic equations describing each situation and then solve the system to find the original number of students in room Q. Given Data / Assumptions: - Let P be the original number of students in room P. - Let Q be the original number of students in room Q. - Scenario 1: When 10 students move from P to Q, the numbers become equal in both rooms. - Scenario 2: When 20 students move from Q to P, the number in P becomes double the number in Q. - No students enter or leave the building; only movement between the two rooms is considered. Concept / Approach: We express each scenario as an equation. In the first scenario, the new population of P is P - 10 and the new population of Q is Q + 10, and these two are equal. In the second scenario, the new population of P is P + 20 and the new population of Q is Q - 20, and now P + 20 is twice Q - 20. Solving these two equations simultaneously gives us P and Q. Finally, we select the option that matches the value of Q. Step-by-Step Solution: Step 1: From scenario 1, when 10 students move from P to Q, we have P - 10 = Q + 10. Step 2: Rearranging gives P - Q = 20. This is our first equation. Step 3: From scenario 2, when 20 students move from Q to P, we have P + 20 = 2(Q - 20), because the number in P becomes double the number in Q. Step 4: Expand the second equation: P + 20 = 2Q - 40, so P - 2Q = -60. Step 5: Now we solve the system: (1) P - Q = 20 and (2) P - 2Q = -60. Step 6: Subtract equation (2) from equation (1): (P - Q) - (P - 2Q) = 20 - (-60) gives Q = 80. Step 7: Substitute Q = 80 into P - Q = 20 to get P - 80 = 20, so P = 100. Verification / Alternative check: Check scenario 1 with P = 100 and Q = 80: after sending 10 from P to Q, room P has 90 and room Q has 90, so they are equal. Check scenario 2: after sending 20 from Q to P, room P has 120 and room Q has 60. Indeed, 120 is twice 60, so the second condition is satisfied as well. This confirms that Q = 80 is correct. Why Other Options Are Wrong: Option 100 or 70 or 120 or 90: Substituting any of these values for Q into the equations does not satisfy both scenarios simultaneously. Either the equal-student condition fails or the double-condition fails, so these values cannot be correct. Common Pitfalls: A common error is mixing up which room gains and which loses students in each scenario, leading to equations with wrong signs. Another mistake is misinterpreting double as P = Q + 20 instead of P = 2Q. Carefully writing updated counts for each room and explicitly forming the equations before solving helps avoid these mistakes. Final Answer: The number of students originally in room Q is 80.

Question 14

On sports day, 12 children are placed in each column, forming 98 columns. Later they are asked to stand in concentric circles so that the innermost circle has 1 student, the next circle has 2 students, the third has 3 students, and so on. How many s…

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Introduction / Context: This puzzle combines arithmetic with the concept of triangular numbers. We are first told how many children there are in total by describing them in columns. Then they are reorganized into concentric circles, with each successive circle having one more student than the previous one. The pattern 1, 2, 3, 4, ... is a sequence whose partial sums give triangular numbers. The question asks how many circles are formed, which is equivalent to finding which triangular number equals the total number of students. Given Data / Assumptions: - There are 12 children in each of 98 columns. - Total number of students is therefore 12 × 98. - In the circle arrangement, the first (innermost) circle has 1 student, the second has 2, the third has 3, and so on up to k circles. Concept / Approach: The total number of children in k such circles is 1 + 2 + 3 + ... + k, which is the k-th triangular number. This sum equals k(k + 1) / 2. We first compute the total number of students from the column information, then set this total equal to k(k + 1) / 2 and solve the resulting quadratic equation for k. The integer solution for k is the required number of circles. Step-by-Step Solution: Step 1: Compute total number of students in columns: total N = 12 × 98. Step 2: 12 × 98 = 12 × (100 - 2) = 1200 - 24 = 1176 students. Step 3: In the circle arrangement, the total number of students equals 1 + 2 + 3 + ... + k, which is k(k + 1) / 2. Step 4: Set k(k + 1) / 2 = 1176 and multiply both sides by 2: k(k + 1) = 2352. Step 5: We need integer k such that k^2 + k - 2352 = 0. Try k = 48: 48 × 49 = 2352, which satisfies the equation. Step 6: Therefore k = 48 circles are formed. Verification / Alternative check: Check by summation formula: for k = 48, the total number of students is 48 × 49 / 2 = 24 × 49 = 1176. This matches the total number of children previously computed from the column arrangement, confirming that all students are exactly used and no students are left out or missing. Thus the pattern of 1, 2, 3, ..., 48 circles correctly accounts for every child. Why Other Options Are Wrong: Options 44, 46, 49: For each of these values of k, k(k + 1) / 2 does not equal 1176. Either too few or too many students would be accounted for, so they cannot be correct. Option None of these: Not correct because 48 matches the requirements exactly. Common Pitfalls: One common mistake is to forget or misapply the triangular number formula and attempt to add numbers manually, which is error-prone for large totals. Another error is to assume that 1176 is close to 50 × 50 / 2 = 1250 and then guess nearby values without checking the exact product. Using the precise equation k(k + 1) = 2352 and testing exact factor pairs avoids such guessing and ensures the correct answer. Final Answer: The children will form 48 concentric circles.

Question 15

The first 10 odd numbers starting from 1 are 1, 3, 5, ..., 19. By what percentage is their average less than the last term 19?

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Introduction / Context: This question deals with averages of arithmetic sequences and percentage comparison. We are given the first ten odd numbers starting from 1 and asked to compare their arithmetic mean with the last term of the sequence, which is 19. The phrase by what percentage is the average less than the last term means we must compute the difference between the last term and the average, then express that difference as a percentage of the last term itself. Given Data / Assumptions: - The first 10 odd numbers starting from 1 form an arithmetic progression: 1, 3, 5, ..., 19. - The first term is 1, the common difference is 2, and the 10th term is 19. - The arithmetic mean of a sequence is (sum of terms) / (number of terms). For an arithmetic progression, the mean equals the average of the first and last terms. - We interpret the required percentage as (difference / last term) × 100. Concept / Approach: We first find the average of the 10 odd numbers using the simple property of arithmetic progressions: mean = (first term + last term) / 2. Then we compute how much smaller this average is than the last term by taking last term minus average. Finally, to express this as a percentage, we divide that difference by the last term and multiply by 100. The result will be a fraction that we can match against the given answer forms. Step-by-Step Solution: Step 1: List the first 10 odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. Step 2: Note that this is an arithmetic progression with first term a = 1 and last term l = 19. Step 3: The average (mean) of an arithmetic progression is (a + l) / 2. Thus mean = (1 + 19) / 2 = 20 / 2 = 10. Step 4: The last term is 19, so the average is less than the last term by 19 - 10 = 9. Step 5: Required percentage = (difference / last term) × 100 = (9 / 19) × 100. Step 6: Write this as a single fraction: 9 × 100 / 19 = 900 / 19 percent. Verification / Alternative check: We can verify by computing the sum of the sequence directly: sum S = n/2 × (first + last) = 10/2 × (1 + 19) = 5 × 20 = 100. The mean is S / n = 100 / 10 = 10, which agrees with the earlier calculation. The last term is still 19, and the difference is 9. Dividing 9 by 19 and multiplying by 100 again yields 900 / 19 percent. This consistency confirms the calculation is correct. Why Other Options Are Wrong: Options 991/16 %, 874/13 %, 719/17 %: None of these fractions simplifies to 9 × 100 / 19. They correspond to very different decimal values and are not obtained by the required difference over the last term. Option None of these: Incorrect, because 900/19 % is exactly one of the options and matches the correct calculation. Common Pitfalls: A common mistake is to compute the percentage with respect to the average instead of the last term, leading to (9 / 10) × 100 = 90 percent. Another error is to misinterpret the question as asking by what percentage the average is greater instead of less, which reverses the sign conceptually. Carefully reading that we want the percentage by which the average is less than the last term and using difference / last term × 100 avoids such confusion. Final Answer: The average is less than the last term by 900/19 percent.

Question 16

In a street there are some houses to be painted using a fixed set of colours. If 4 houses are painted with each colour, exactly 1 house remains unpainted. If instead 5 houses are painted with each colour, exactly 1 colour remains unused. How many ho…

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Introduction / Context: This is a combinational arithmetic reasoning puzzle involving houses and colours. We are told that the same set of colours is used in two different ways. In one scheme, houses are grouped by colour leaving one house unpainted. In the other scheme, houses are grouped differently, leaving one colour unused. By interpreting these conditions algebraically, we can determine how many houses are on the street. Given Data / Assumptions: - Let H be the total number of houses. - Let C be the total number of colours available. - Scenario 1: If 4 houses are painted with each colour, one house is left unpainted. - Scenario 2: If 5 houses are painted with each colour, exactly one colour is not used at all. - Each house receives exactly one colour and each used colour is applied to the stated number of houses in that scenario. Concept / Approach: We translate each scenario into an equation relating H and C. In the first scenario, painting 4 houses per colour for all C colours covers 4C houses and leaves 1 unpainted, so H = 4C + 1. In the second scenario, one colour stays unused, so only C - 1 colours are used with 5 houses each, covering 5(C - 1) houses; this must equal the total H. Equating these two expressions for H gives a linear equation in C, from which we can find C and then H. Finally, we match H with the correct option. Step-by-Step Solution: Step 1: From scenario 1, 4 houses per colour and one house left unpainted imply H = 4C + 1. Step 2: From scenario 2, 5 houses per colour with one unused colour imply that only C - 1 colours are used, so H = 5(C - 1). Step 3: Equate the two expressions for H: 4C + 1 = 5(C - 1). Step 4: Expand the right-hand side: 5(C - 1) = 5C - 5, so we have 4C + 1 = 5C - 5. Step 5: Rearrange: 4C + 1 - 5C + 5 = 0 leads to -C + 6 = 0, so C = 6. Step 6: Substitute C = 6 into H = 4C + 1 to obtain H = 4 × 6 + 1 = 24 + 1 = 25 houses. Verification / Alternative check: Check scenario 1 with H = 25 and C = 6: 4 houses per colour uses 4 × 6 = 24 houses, leaving exactly 1 house unpainted, as required. Check scenario 2: using 5 houses per colour but only C - 1 = 5 colours gives 5 × 5 = 25 houses, so one colour is unused and all houses are painted. Both conditions are satisfied, confirming that 25 houses is correct. Why Other Options Are Wrong: Option 6 and 7: These are far too small to accommodate the described painting schemes; they do not fit both equations simultaneously. Option 32: If H = 32, then from the first scenario C would be (32 - 1) / 4 = 31 / 4, not an integer, contradicting the notion of an integer number of colours. Option None of these: Not correct, because 25 houses perfectly satisfy both scenarios. Common Pitfalls: Learners often confuse houses left unpainted with colours left unused, or misread the second condition as leaving one house unpainted again instead of leaving a colour unused. Another mistake is assuming that the same number of colours is used in both cases without adjusting for the unused colour. Writing clear equations for each scenario before solving helps avoid such confusion. Final Answer: There are 25 houses in the street.

Question 17

Five bags are arranged in order. The first bag contains some fruits. Each subsequent bag contains one quarter as many fruits as the previous bag, and the fifth (last) bag contains 4 fruits. How many fruits are in the first bag?

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Introduction / Context: This is a geometric progression type problem disguised in a word puzzle about bags of fruits. We are told that each bag has one quarter as many fruits as the previous bag, and the last bag in a sequence of five bags contains 4 fruits. Because each bag is obtained from the previous one by multiplying by a fixed ratio, the numbers of fruits form a geometric sequence. The question asks us to determine how many fruits were initially in the first bag. Given Data / Assumptions: - There are 5 bags, labeled from first to fifth in order. - Let the number of fruits in the first bag be F. - Each subsequent bag has one quarter as many fruits as the previous one, so the common ratio is 1/4. - The fifth bag contains 4 fruits. - All fruit counts are assumed to be whole numbers. Concept / Approach: Since each bag contains 1/4 of the fruits of the previous bag, the sequence of fruit counts is a geometric progression: F, F/4, F/16, F/64, F/256. The fifth term is given as 4, so we can set F/256 = 4 and solve for F. This is a straightforward algebraic manipulation using the idea that dividing by 1/4 repeatedly is equivalent to dividing by 4 each step, or multiplying by 4 when moving backward in the sequence. Step-by-Step Solution: Step 1: Write the fruit counts for the bags as a geometric sequence. First bag: F, Second bag: F/4, Third bag: F/4^2 = F/16, Fourth bag: F/4^3 = F/64, Fifth bag: F/4^4 = F/256. Step 2: We are told that the fifth bag has 4 fruits, so F/256 = 4. Step 3: Multiply both sides of the equation by 256: F = 4 × 256. Step 4: Compute 4 × 256 = 1024. Step 5: Hence the first bag contains F = 1024 fruits. Verification / Alternative check: We can check by reconstructing the entire sequence: First bag: 1024 fruits, Second bag: 1024/4 = 256 fruits, Third bag: 256/4 = 64 fruits, Fourth bag: 64/4 = 16 fruits, Fifth bag: 16/4 = 4 fruits. This exactly matches the condition that each bag has one quarter of the previous and that the last bag contains 4 fruits. Why Other Options Are Wrong: Option 2596, 256, 64: Substituting any of these as the first bag's count and successively dividing by 4 will not yield 4 fruits in the fifth bag. Option None of these: Incorrect because 1024 is one of the options and fits the requirements perfectly. Common Pitfalls: A frequent misunderstanding is to interpret the phrase reduced by one quarter as meaning subtracting a fixed amount instead of multiplying by 1/4. In this problem, however, the context of each bag having one quarter as many fruits clearly indicates a multiplicative relationship. Recognizing the difference between additive and multiplicative change is crucial for solving such problems correctly. Final Answer: The first bag contains 1024 fruits.

Question 18

In an arithmetic progression, the 18th term is 29 and the 29th term is 18. What is the value of the 49th term of this progression?

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Introduction / Context: This question uses standard properties of arithmetic progressions (APs). In an AP, the difference between consecutive terms is constant and is called the common difference. If we know any two terms and their positions in the sequence, we can find both the first term and the common difference. Once these are known, any other term can be computed using the general term formula. Here we are given the 18th and 29th terms and asked to find the 49th term. Given Data / Assumptions: - The sequence is an arithmetic progression. - Let a be the first term and d be the common difference. - The 18th term is 29, so a + 17d = 29. - The 29th term is 18, so a + 28d = 18. - We want the 49th term, which is a + 48d. Concept / Approach: We use the nth term formula for an arithmetic progression: T_n = a + (n - 1)d. By plugging in n = 18 and n = 29, we obtain two linear equations in a and d. Solving this system gives us the values of a and d. Then we substitute these into T_49 = a + 48d to get the required term. This method is general and works for any pair of known terms in an AP. Step-by-Step Solution: Step 1: Use the 18th term: T_18 = a + 17d = 29. Step 2: Use the 29th term: T_29 = a + 28d = 18. Step 3: Subtract the first equation from the second: (a + 28d) - (a + 17d) = 18 - 29. Step 4: This simplifies to 11d = -11, so d = -1. Step 5: Substitute d = -1 back into a + 17d = 29 to find a: a + 17(-1) = 29, so a - 17 = 29 and a = 46. Step 6: The 49th term is T_49 = a + 48d = 46 + 48(-1) = 46 - 48 = -2. Verification / Alternative check: We can check the given terms using a = 46 and d = -1. The 18th term is a + 17d = 46 + 17(-1) = 46 - 17 = 29, which matches the given value. The 29th term is a + 28d = 46 + 28(-1) = 46 - 28 = 18, which also matches. This confirms that the parameters of the AP are correct and that T_49 = -2 is reliable. Why Other Options Are Wrong: Options -3, 0, 1: None of these values result from the correct substitution into T_49. They correspond to incorrect values of d or miscalculations of the term index. Option None of these: Incorrect because -2 is one of the listed options and is the correct 49th term. Common Pitfalls: A typical mistake is to mix up the (n - 1) factor and accidentally use n instead, which shifts the index by one and produces wrong terms. Another error is in subtracting the equations: if the signs are mishandled, a wrong common difference is obtained. Being careful with algebraic manipulation and using the standard T_n formula correctly avoids these problems. Final Answer: The 49th term of the arithmetic progression is -2.

Question 19

In basic geometry, two angles whose sum is 180 degrees are called what kind of angles?

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Introduction / Context: This question tests knowledge of standard terminology for angle relationships in geometry. Many questions in geometry, trigonometry, and coordinate geometry implicitly assume familiarity with terms like complementary, supplementary, and alternate angles. Knowing the definitions enables quick recognition of relationships and simplifies problem solving involving straight lines and right angles. Given Data / Assumptions: - We consider two angles whose measures add up to 180 degrees. - These angles may be adjacent (forming a straight line) or separate, but their measures still sum to 180 degrees. - We use conventional Euclidean geometry terminology. Concept / Approach: In geometry, two angles whose sum is 90 degrees are called complementary angles. Two angles whose sum is 180 degrees are called supplementary angles. Alternate angles are formed when a transversal intersects two lines and refer to relative positions, not necessarily to any particular sum. Vertical angles are formed by intersecting lines and are equal to each other, again without specifying any fixed sum. Therefore, we identify the correct name by matching the defining property: sum equals 180 degrees. Step-by-Step Solution: Step 1: Recall that complementary angles are defined as two angles whose measures add up to 90 degrees. Step 2: Recall that supplementary angles are defined as two angles whose measures add up to 180 degrees. Step 3: Alternate angles arise when a transversal cuts two lines and have positional descriptions such as alternate interior or alternate exterior; they are not defined by a fixed sum of 180 degrees. Step 4: Vertical angles are opposite angles formed when two straight lines intersect, and they are equal in measure, again not defined by sum 180 degrees. Step 5: Since the question states that the sum is 180 degrees, by definition these angles are supplementary angles. Verification / Alternative check: Consider a straight line. Any two adjacent angles forming a straight line sum to 180 degrees and are typically cited as examples of supplementary angles. For instance, angles of 110 degrees and 70 degrees sum to 180 degrees and are therefore supplementary. No other standard name is used for this specific sum, confirming that supplementary is the correct term. Why Other Options Are Wrong: Option Complementary angles: They sum to 90 degrees, not 180 degrees. Option Alternate angles: This term describes relative positions with respect to a transversal and does not specify any fixed sum. Option None of the above: Incorrect because the standard name supplementary angles exactly matches the given property. Option Vertical angles: These are equal angles formed at the intersection of two lines, with no requirement that they sum to 180 degrees. Common Pitfalls: Students often confuse complementary and supplementary angles, mixing up which sum corresponds to 90 degrees and which to 180 degrees. A useful memory aid is that co in complementary goes with corner and right angles (90 degrees), while sup in supplementary can be linked with straight (180 degrees). Remembering this helps avoid mislabeling angle relationships. Final Answer: Two angles whose sum is 180 degrees are called supplementary angles.

Question 20

In computing, the binary number system is based on which powers of a single base?

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Introduction / Context: This question tests knowledge of number systems used in computing. Binary is the foundational number system for digital electronics and computer architecture. Understanding which base underlies binary helps in converting between number systems, understanding bit representations, and working with low-level programming or digital logic design. Given Data / Assumptions: - The binary system uses only two digits: 0 and 1. - Positional number systems represent numbers as sums of powers of a base multiplied by digits. - We need to identify which base's powers are used in binary representation. Concept / Approach: In any positional number system, a numeral d_k d_{k-1} ... d_1 d_0 represents the value Σ d_i × base^i, where the base is fixed and the digits d_i are constrained by the base. For decimal, the base is 10; for binary, the digits allowed are 0 and 1. Since there are two distinct digit values, the base must be 2. Every binary digit (bit) represents a power of 2, starting from 2^0 at the rightmost bit. Step-by-Step Solution: Step 1: Recall that in decimal (our everyday system), we use digits 0 through 9, and each position represents a power of 10 (ones, tens, hundreds, etc.). Step 2: In binary, only digits 0 and 1 are allowed. Therefore the base must be 2, because we have exactly two digit symbols. Step 3: A binary number like 1011₂ represents 1 × 2^3 + 0 × 2^2 + 1 × 2^1 + 1 × 2^0. Step 4: All place values in binary are powers of 2: 1, 2, 4, 8, 16, and so on. Hence the system uses powers of 2. Step 5: Therefore the correct choice is that the binary system uses powers of 2. Verification / Alternative check: Consider 13 in decimal, which is 1101 in binary. We can verify: 1 × 2^3 = 8, 1 × 2^2 = 4, 0 × 2^1 = 0, 1 × 2^0 = 1. Summing these gives 8 + 4 + 0 + 1 = 13, showing explicitly that binary digits weigh powers of 2. This reinforces that the base is 2, not 0, 1, or 10. Why Other Options Are Wrong: Option 0: A base of zero would not allow meaningful positional representation and is not used in positional number systems. Option 1: Using base 1 would collapse all positional distinctions and cannot represent general integers in a standard positional system. Option 10: This corresponds to the decimal system, not the binary system. Option None of these: Incorrect because powers of 2 are exactly the right choice and are listed as an option. Common Pitfalls: A frequent confusion is between the symbols used and the base. Some learners think that because the digits are 0 and 1, the base must be 1, which is incorrect. The base counts how many distinct digit symbols exist, not the maximum digit value itself. In binary there are 2 distinct digits, 0 and 1, so the base is 2 and the positional values are powers of 2. Final Answer: The binary number system uses powers of 2.

Arithmetic Reasoning Questions