Q: Using the standard order of operations (multiplication and division left to right, then addition), evaluate the expression: 5 + 2 * 10 ÷ 2.

Introduction: This question checks understanding of order of operations. In arithmetic, multiplication and division are performed before addition and subtraction. Also, multiplication and division have the same priority, so they are carried out from left to right. Therefore, in 5 + 2 * 10 ÷ 2, we first compute 2 * 10, then divide by 2, and only then add 5. Getting this wrong usually comes from adding 5 too early or grouping division incorrectly. Given Data / Assumptions: Expression: 5 + 2 * 10 ÷ 2 Rule: perform * and ÷ from left to right Then perform + Concept / Approach: Evaluate the multiplication and division part first. Since 2 * 10 ÷ 2 is a chain, compute left to right: (2 * 10) ÷ 2. Finally add 5 to the resulting value. This is the standard BODMAS/PEMDAS logic (with the important note that division is not done before multiplication; it is left to right). Step-by-Step Solution: Start: 5 + 2 * 10 ÷ 2 Compute multiplication: 2 * 10 = 20 Now divide: 20 ÷ 2 = 10 Now add: 5 + 10 = 15 Verification / Alternative check: You can rewrite the expression as 5 + (2 * 10 / 2). Since 10/2 = 5, the inside becomes 2*5 = 10. Then 5 + 10 = 15. Same result confirms correctness. Why Other Options Are Wrong: 10 would be chosen if the +5 is ignored. 20 might come from doing 5 + 2 first and then multiplying, which is not allowed without parentheses. 35 might come from 5 + 2*10 = 25 and then adding 10 again incorrectly. 25 is from stopping after 5 + 20 without dividing by 2. Common Pitfalls: Adding before completing multiplication/division, or assuming division must be done before multiplication instead of left-to-right evaluation. Final Answer: The value of 5 + 2 * 10 ÷ 2 is 15.

Q: For all real values of x, the expression 5 sin(x) + 12 cos(x) + r is required to be greater than or equal to 0. What is the smallest value of r that satisfies this condition?

Introduction: This question tests the minimum/maximum value of a trigonometric linear combination of sin(x) and cos(x). Expressions of the form A sin(x) + B cos(x) have a fixed amplitude equal to sqrt(A^2 + B^2). That means the expression varies between -sqrt(A^2 + B^2) and +sqrt(A^2 + B^2) as x changes. To ensure 5 sin(x) + 12 cos(x) + r is always at least 0, r must shift the entire range upward so that the minimum becomes 0. Finding the smallest such shift gives the required minimum r. Given Data / Assumptions: Expression: 5 sin(x) + 12 cos(x) + r Must satisfy: 5 sin(x) + 12 cos(x) + r ≥ 0 for all real x Amplitude rule: A sin(x) + B cos(x) ranges from -sqrt(A^2 + B^2) to +sqrt(A^2 + B^2) Concept / Approach: Compute the minimum of 5 sin(x) + 12 cos(x). It is -sqrt(5^2 + 12^2). Then require: minimum + r ≥ 0, so r ≥ sqrt(5^2 + 12^2). Choose the smallest r that satisfies this, which is exactly that amplitude value. Step-by-Step Solution: A = 5 and B = 12 Amplitude = sqrt(A^2 + B^2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 Minimum of 5 sin(x) + 12 cos(x) = -13 We need -13 + r ≥ 0 for all x So r ≥ 13 Smallest such value is r = 13 Verification / Alternative check: If r = 13, the expression becomes 5 sin(x) + 12 cos(x) + 13. Since 5 sin(x) + 12 cos(x) is at least -13, the total is at least 0. If r were even slightly less than 13, then at the x where 5 sin(x) + 12 cos(x) reaches -13, the expression would become negative, violating the requirement. Why Other Options Are Wrong: -13 and -11 shift the expression downward, making it negative for many x values. 0 and 11 are not enough to offset the minimum -13, so the expression would still go below 0. Only 13 ensures the minimum reaches exactly 0 without being negative. Common Pitfalls: Assuming sin and cos can both be 1 at the same time, adding coefficients as 5+12, or forgetting to take the square root when computing amplitude. Final Answer: The smallest value of r is 13.

Question 1

If a - b = 3 and a^2 + b^2 = 29 for real numbers a and b, what is the value of the product ab?

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Introduction: This question tests using algebraic identities to connect a - b, a^2 + b^2, and ab. The key identity is (a - b)^2 = a^2 + b^2 - 2ab. Since (a - b) and a^2 + b^2 are both given, we can compute (a - b)^2 and then solve for ab. This avoids solving for a and b individually, which would be longer and unnecessary. It is a common simplification technique in algebra and aptitude exams. Given Data / Assumptions: a - b = 3 a^2 + b^2 = 29 We need to find ab Identity used: (a - b)^2 = a^2 + b^2 - 2ab Concept / Approach: Square the difference to get (a - b)^2. Then substitute into the identity (a - b)^2 = a^2 + b^2 - 2ab. With a^2 + b^2 known, the only unknown becomes ab. Solve the resulting linear equation for ab. Step-by-Step Solution: Given a - b = 3, so (a - b)^2 = 3^2 = 9 Use identity: (a - b)^2 = a^2 + b^2 - 2ab Substitute values: 9 = 29 - 2ab 2ab = 29 - 9 = 20 ab = 20/2 = 10 Verification / Alternative check: We can also verify consistency by finding possible a and b. From a - b = 3, let a = b + 3. Then a^2 + b^2 = (b+3)^2 + b^2 = 2b^2 + 6b + 9 = 29. So 2b^2 + 6b - 20 = 0 => b^2 + 3b - 10 = 0 => (b+5)(b-2)=0, so b=2 or b=-5. Corresponding a=5 or a=-2. In both cases, ab = 10. Verified. Why Other Options Are Wrong: Any value other than 10 would violate the identity relation because (a - b)^2 and a^2 + b^2 are fixed. The equation 9 = 29 - 2ab forces ab to be 10 exactly. Common Pitfalls: Using (a + b)^2 instead of (a - b)^2, forgetting the -2ab term, or mis-subtracting 29 - 9. Final Answer: The value of ab is 10.

Question 2

If x = 1 - q and y = 2q + 1, then for what value of q are x and y equal (that is, x = y)?

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Introduction: This question tests solving a simple linear equation formed by equating two expressions. Since x and y are defined in terms of q, the condition “x and y are equal” means substitute the definitions and set them equal: 1 - q = 2q + 1. Then solve for q by collecting like terms. This is a direct single-step algebra problem once the equality is written correctly. Given Data / Assumptions: x = 1 - q y = 2q + 1 Condition: x = y q is a real number Concept / Approach: Set 1 - q equal to 2q + 1, then move all q terms to one side and constants to the other. Solve for q. Because both sides contain a constant 1, they cancel neatly, making the solution immediate. Step-by-Step Solution: Set x = y: 1 - q = 2q + 1 Subtract 1 from both sides: -q = 2q Add q to both sides: 0 = 3q So q = 0 Verification / Alternative check: If q = 0, then x = 1 - 0 = 1 and y = 2*0 + 1 = 1. Since x = y, the solution is correct. If q is not 0, then -q and 2q cannot be equal, so equality fails. Why Other Options Are Wrong: q = 1 gives x = 0 and y = 3 (not equal). q = -1 gives x = 2 and y = -1 (not equal). q = -2 gives x = 3 and y = -3 (not equal). q = 2 gives x = -1 and y = 5 (not equal). Only 0 makes x and y match. Common Pitfalls: Changing signs incorrectly when moving -q, or mistakenly setting 1 - q = 2q - 1 (wrong expression for y). Final Answer: The value of q is 0.

Question 3

Evaluate the expression 18800 ÷ 470 ÷ 20, where division is performed strictly from left to right. What is the final value?

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Introduction: This problem tests correct handling of chained division. A common mistake is to treat a ÷ b ÷ c as a ÷ (b ÷ c), but standard order of operations says multiplication and division are performed from left to right when no parentheses are present. So we evaluate 18800 ÷ 470 first, then divide the result by 20. Recognizing the left-to-right rule is the main point. Given Data / Assumptions: Expression: 18800 ÷ 470 ÷ 20 Division is performed from left to right (as stated) No parentheses, so we do sequential division Concept / Approach: Compute (18800 ÷ 470) first. Then take that quotient and divide by 20. If possible, simplify using exact division: 470 goes into 18800 evenly because 47*4 = 188, so 470*40 = 18800. This makes the calculation fast and exact. Step-by-Step Solution: Evaluate left to right: (18800 ÷ 470) ÷ 20 18800 ÷ 470 = 40 (because 470 * 40 = 18800) Now divide by 20: 40 ÷ 20 = 2 Verification / Alternative check: You can rewrite the expression as 18800 / (470 * 20) only if you keep left-to-right logic: (18800/470)/20 = 18800/(470*20). That equals 18800/9400 = 2, confirming the same result. Why Other Options Are Wrong: 1 would occur if the last step were mistakenly 40 ÷ 40. 3 or 4 are typical guesses from rounding 18800/470 incorrectly. 0.5 could come from mistakenly doing 18800 ÷ (470 ÷ 20), which is wrong because it changes the grouping. Common Pitfalls: Grouping division incorrectly, doing 470 ÷ 20 first, or confusing left-to-right rule for division/multiplication. Final Answer: The value of the expression is 2.

Question 4

Approximate the value of (50.003% of 99.8) ÷ 49.988 using sensible estimation. Which of the following is the closest approximate value?

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Introduction: This question tests approximation and estimation with percentages and decimals. The expression looks complex, but the numbers are deliberately chosen to simplify: 50.003% is extremely close to 50%, and 99.8 is close to 100. Also, the denominator 49.988 is close to 50. So the overall expression should be close to (50% of 100) / 50 = 50 / 50 = 1. The goal is not exact calculation but picking the closest option quickly and confidently. Given Data / Assumptions: Expression: (50.003% of 99.8) ÷ 49.988 50.003% = 50.003/100 ≈ 0.50003 99.8 ≈ 100 and 49.988 ≈ 50 for estimation We choose the closest option (approximate answer) Concept / Approach: Convert percent to a multiplier and estimate stepwise. First estimate the “percent of” part, then divide by the denominator. Use closeness: 50.003% is almost half, and 99.8 is almost 100, so numerator is almost 50. Denominator is almost 50, so result is almost 1. We can also do a slightly tighter estimate to confirm it is still closest to 1. Step-by-Step Solution: 50.003% = 0.50003 (approximately) Numerator ≈ 0.50003 * 99.8 0.5 * 99.8 = 49.9 Extra part: 0.00003 * 99.8 ≈ 0.002994 So numerator ≈ 49.902994 Denominator = 49.988 ≈ 50 Result ≈ 49.903 / 49.988 ≈ 0.998 (very close to 1) Verification / Alternative check: Using pure rounding: (50% of 100) / 50 = 1. Since our refined estimate is about 0.998, the nearest option remains 1, not 0 or 2. Why Other Options Are Wrong: 0 is impossible because all quantities are positive. 2 would require numerator about twice the denominator (~100), which it is not. 50 is far too large and ignores the division. 0.5 would require numerator about half the denominator (~25), which is not the case. Common Pitfalls: Treating 50.003% as 50.003 (not dividing by 100), or rounding too aggressively without checking whether the result stays near 1. Final Answer: The closest approximate value is 1.

Question 5

If ab = 45 and bc = 60 for nonzero real numbers a, b, and c, find the value of: (c^2 - a^2) / (c^2 + a^2). Choose the correct simplified value.

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Introduction: This question tests algebraic manipulation using given products. Even though a, b, and c are not individually known, ab and bc allow you to express a and c in terms of b. Since the expression involves a^2 and c^2, representing both as something divided by b^2 helps cancel b. The key insight is that the expression (c^2 - a^2)/(c^2 + a^2) depends only on the ratio of c^2 and a^2, and b will cancel completely. Given Data / Assumptions: ab = 45 bc = 60 b ≠ 0 (so we can divide by b) We need (c^2 - a^2) / (c^2 + a^2) Concept / Approach: From ab = 45, write a = 45/b. From bc = 60, write c = 60/b. Then compute a^2 and c^2. Substitute into the expression. Because both a^2 and c^2 contain 1/b^2, it cancels out, leaving a simple fraction involving 45^2 and 60^2. Step-by-Step Solution: a = 45/b and c = 60/b a^2 = 45^2 / b^2 = 2025 / b^2 c^2 = 60^2 / b^2 = 3600 / b^2 (c^2 - a^2) / (c^2 + a^2) = (3600/b^2 - 2025/b^2) / (3600/b^2 + 2025/b^2) = (3600 - 2025) / (3600 + 2025) = 1575 / 5625 Divide numerator and denominator by 225: 1575/225 = 7 and 5625/225 = 25 So value = 7/25 Verification / Alternative check: Pick a convenient b, say b = 15. Then a = 45/15 = 3 and c = 60/15 = 4. Expression becomes (16 - 9)/(16 + 9) = 7/25, confirming the result. Why Other Options Are Wrong: 7/5 and 7 are too large (the fraction must be less than 1 in magnitude because |c^2 - a^2| < c^2 + a^2 for positive squares). 7/125 is too small. 5/7 is a different ratio and does not match substitution. Common Pitfalls: Trying to solve for a, b, c uniquely (not possible), forgetting to square, or failing to cancel b^2 correctly when combining fractions. Final Answer: The value is 7/25.

Question 6

Using the standard order of operations (multiplication and division left to right, then addition), evaluate the expression: 5 + 2 * 10 ÷ 2.

Accepted Answer

Introduction: This question checks understanding of order of operations. In arithmetic, multiplication and division are performed before addition and subtraction. Also, multiplication and division have the same priority, so they are carried out from left to right. Therefore, in 5 + 2 * 10 ÷ 2, we first compute 2 * 10, then divide by 2, and only then add 5. Getting this wrong usually comes from adding 5 too early or grouping division incorrectly. Given Data / Assumptions: Expression: 5 + 2 * 10 ÷ 2 Rule: perform * and ÷ from left to right Then perform + Concept / Approach: Evaluate the multiplication and division part first. Since 2 * 10 ÷ 2 is a chain, compute left to right: (2 * 10) ÷ 2. Finally add 5 to the resulting value. This is the standard BODMAS/PEMDAS logic (with the important note that division is not done before multiplication; it is left to right). Step-by-Step Solution: Start: 5 + 2 * 10 ÷ 2 Compute multiplication: 2 * 10 = 20 Now divide: 20 ÷ 2 = 10 Now add: 5 + 10 = 15 Verification / Alternative check: You can rewrite the expression as 5 + (2 * 10 / 2). Since 10/2 = 5, the inside becomes 2*5 = 10. Then 5 + 10 = 15. Same result confirms correctness. Why Other Options Are Wrong: 10 would be chosen if the +5 is ignored. 20 might come from doing 5 + 2 first and then multiplying, which is not allowed without parentheses. 35 might come from 5 + 2*10 = 25 and then adding 10 again incorrectly. 25 is from stopping after 5 + 20 without dividing by 2. Common Pitfalls: Adding before completing multiplication/division, or assuming division must be done before multiplication instead of left-to-right evaluation. Final Answer: The value of 5 + 2 * 10 ÷ 2 is 15.

Question 7

For all real values of x, the expression 5 sin(x) + 12 cos(x) + r is required to be greater than or equal to 0. What is the smallest value of r that satisfies this condition?

Accepted Answer

Introduction: This question tests the minimum/maximum value of a trigonometric linear combination of sin(x) and cos(x). Expressions of the form A sin(x) + B cos(x) have a fixed amplitude equal to sqrt(A^2 + B^2). That means the expression varies between -sqrt(A^2 + B^2) and +sqrt(A^2 + B^2) as x changes. To ensure 5 sin(x) + 12 cos(x) + r is always at least 0, r must shift the entire range upward so that the minimum becomes 0. Finding the smallest such shift gives the required minimum r. Given Data / Assumptions: Expression: 5 sin(x) + 12 cos(x) + r Must satisfy: 5 sin(x) + 12 cos(x) + r ≥ 0 for all real x Amplitude rule: A sin(x) + B cos(x) ranges from -sqrt(A^2 + B^2) to +sqrt(A^2 + B^2) Concept / Approach: Compute the minimum of 5 sin(x) + 12 cos(x). It is -sqrt(5^2 + 12^2). Then require: minimum + r ≥ 0, so r ≥ sqrt(5^2 + 12^2). Choose the smallest r that satisfies this, which is exactly that amplitude value. Step-by-Step Solution: A = 5 and B = 12 Amplitude = sqrt(A^2 + B^2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 Minimum of 5 sin(x) + 12 cos(x) = -13 We need -13 + r ≥ 0 for all x So r ≥ 13 Smallest such value is r = 13 Verification / Alternative check: If r = 13, the expression becomes 5 sin(x) + 12 cos(x) + 13. Since 5 sin(x) + 12 cos(x) is at least -13, the total is at least 0. If r were even slightly less than 13, then at the x where 5 sin(x) + 12 cos(x) reaches -13, the expression would become negative, violating the requirement. Why Other Options Are Wrong: -13 and -11 shift the expression downward, making it negative for many x values. 0 and 11 are not enough to offset the minimum -13, so the expression would still go below 0. Only 13 ensures the minimum reaches exactly 0 without being negative. Common Pitfalls: Assuming sin and cos can both be 1 at the same time, adding coefficients as 5+12, or forgetting to take the square root when computing amplitude. Final Answer: The smallest value of r is 13.

Question 8

In a geometric spreading process, a secret can be told by one person to only 3 other persons in 3 minutes, and in each subsequent 3-minute interval every person who already knows the secret tells it to 3 new persons only once in the next interval. I…

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Introduction / Context: This question tests exponential growth and geometric progression in a word problem setting. A secret spreads from person to person, and each person passes the secret to a fixed number of new persons in fixed time intervals. Many aptitude tests use this type of problem to check whether candidates understand repeated multiplication rather than simple addition. The twist here is that every person tells the secret to exactly 3 new persons only once, during the 3-minute interval immediately after they first learn the secret. We must carefully track how many new people are informed during each 3-minute block over a total of 30 minutes and then exclude the original person from the final count. Given Data / Assumptions: - One original person initially knows the secret. - In the first 3 minutes, that person tells the secret to 3 new persons. - In every subsequent 3-minute interval, each person who already knows the secret tells it to exactly 3 new persons only once in the following interval. - Total time is 30 minutes, which equals 10 intervals of 3 minutes each. - We must count all persons who have been told the secret, excluding the original person. Concept / Approach: The number of people who learn the secret in each interval follows a geometric progression with common ratio 3. In interval 1, there are 3 new people. In interval 2, the 3 new people from interval 1 each tell 3 new people, giving 3^2 new people, and so on. After n intervals, the number of new people in interval n is 3^n. The cumulative number of people who have been told (excluding the original person) is the sum of this geometric series from n = 1 to n = 10. Step-by-Step Solution: Interval length = 3 minutes, total time = 30 minutes, so number of intervals = 30 / 3 = 10. New people told in interval 1 = 3^1 = 3. New people told in interval 2 = 3^2 = 9. Generally, in interval n, new people told = 3^n. Total people told (excluding original) = 3^1 + 3^2 + ... + 3^10. This is a geometric series: sum = 3 * (3^10 - 1) / (3 - 1). Compute 3^10 = 59049, so sum = 3 * (59049 - 1) / 2 = 3 * 59048 / 2. 3 * 59048 = 177144; divide by 2 gives 88572. We already excluded the original person in this sum, so the answer is 88572. Verification / Alternative check: We can quickly check reasonableness by approximating growth. Numbers progress as 3, 9, 27, 81, 243 and so on, reaching tens of thousands by the tenth step. Therefore, an answer in the range of about 80,000–100,000 is reasonable. 88572 fits this scale and matches the exact formula for the geometric sum, so the value is consistent both algebraically and intuitively. Why Other Options Are Wrong: Option 77854 is lower than the correct geometric sum and does not match any standard partial sum for 10 intervals. Option 99584 is higher than the correct value 88572 and again does not correspond to the geometric series sum. Option 55654 is far from the correct total and is likely included as a distractor that might come from arithmetic errors in summation. Common Pitfalls: A common mistake is to treat the process as simple addition, for example adding 3 every interval instead of allowing each informed person to tell 3 more people. Another frequent error is to assume each person continues to tell the secret in every interval, which would lead to a different and much larger progression. Some candidates also forget that the question asks for the count excluding the original person, but in our series we never included that person in the sum, so the issue is avoided. Final Answer: Thus, the number of persons who have been told the secret, excluding the original person, is 88572.

Question 9

A tailor has 37.5 metres of cloth and from each full metre he cuts 8 equal pieces of fabric; if he uses exactly half of his cloth for cutting, how many individual pieces can he make altogether from that half portion?

Accepted Answer

Introduction / Context: This problem checks basic proportional reasoning and unit handling. A tailor cuts cloth into equal pieces, and we are told how many pieces come from one metre. We then restrict the amount of cloth actually used to half of the total stock. The question asks for the total number of pieces that can be cut from that used portion. Such questions appear frequently in aptitude tests to see whether candidates can translate words into simple arithmetic based on ratios and fractions. Given Data / Assumptions: - Total cloth length = 37.5 metres. - From every 1 metre of cloth, the tailor cuts 8 equal pieces. - Only half of the total cloth is used for cutting pieces. - The remaining half is not used and therefore does not contribute pieces. - All pieces are of equal size throughout the cutting process. Concept / Approach: First, we find the length of cloth that is actually used, which is half of 37.5 metres. Next, the size of each piece is determined from the information that 1 metre produces 8 pieces, so each piece corresponds to 1/8 of a metre. Finally, we divide the used length of cloth by the length per piece to get the total number of pieces. Straightforward multiplication and division are enough to handle this problem. Step-by-Step Solution: Total cloth = 37.5 metres. Half of the cloth used = 37.5 / 2 = 18.75 metres. From 1 metre, the tailor cuts 8 equal pieces. Length of each piece = 1 / 8 metre. Number of pieces from 18.75 metres = 18.75 / (1 / 8). This equals 18.75 * 8. 18.75 * 8 = 150. Therefore, the tailor can cut 150 pieces from half of the cloth. Verification / Alternative check: An alternative way is to see how many pieces come from 37.5 metres and then halve that count. From 37.5 metres, the total pieces are 37.5 * 8 = 300. Since only half the cloth is used, the number of pieces is 300 / 2 = 150, which confirms the previous result. Why Other Options Are Wrong: Option 300 corresponds to using the entire 37.5 metres, not half of it. Option 175 does not match any consistent calculation involving 8 pieces per metre and half of 37.5 metres. Option 200 would come from an incorrect assumption about the number of pieces per metre or the portion of cloth used. Common Pitfalls: Candidates sometimes forget that only half of the cloth is used and multiply the full length by 8. Others may misinterpret “8 equal pieces from a metre” and incorrectly scale the number of pieces. Another error is mishandling the decimal 37.5 when finding half, so careful arithmetic is necessary. Final Answer: The number of individual pieces the tailor can make from half of the cloth is 150.

Question 10

When the fraction 64/12 is reduced completely to its simplest form by cancelling all common factors of the numerator and denominator, what is the resulting simplified fraction?

Accepted Answer

Introduction / Context: This question assesses basic fraction simplification, an essential arithmetic skill. A fraction is in simplest form when the numerator and denominator share no common factor greater than 1. Here, we are given 64/12 and asked to reduce it fully. Many aptitude tests use such questions to check comfort with highest common factor ideas and cancellation of common factors. Given Data / Assumptions: - Original fraction = 64/12. - We are asked to express it in simplest form. - Simplest form means numerator and denominator are coprime (no common factor beyond 1). - Only positive fractions are considered here, so we do not worry about signs. Concept / Approach: To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and denominator and divide both by that value. For 64 and 12, we identify the largest integer that divides both numbers exactly. Factoring or repeated division by common primes such as 2 can help us quickly reach the simplified fraction. Step-by-Step Solution: Start with 64/12. Both 64 and 12 are even, so divide numerator and denominator by 2. 64 / 2 = 32, 12 / 2 = 6, giving 32/6. 32 and 6 are still even, so divide again by 2. 32 / 2 = 16, 6 / 2 = 3, giving 16/3. Now 16 and 3 have no common factor greater than 1, so 16/3 is in simplest form. Verification / Alternative check: We can use prime factors. 64 = 2^6 and 12 = 2^2 * 3. Cancelling 2^2 from numerator and denominator leaves 2^4 / 3 which equals 16/3. This matches the stepwise simplification and confirms our answer. Why Other Options Are Wrong: Option 19/6 does not equal 64/12 when converted to decimals or compared via cross multiplication. Option 32/4 simplifies further to 8, so it is not equal to 64/12 and is not in simplest form for this fraction. Option “None of the above” is incorrect because we have identified a correct simplified fraction among the choices, namely 16/3. Common Pitfalls: Some learners stop simplification after only one division step, such as 64/12 to 32/6, and forget to reduce further. Others might divide numerator and denominator by a wrong common factor that does not divide both numbers. Remember to keep checking for common factors until no further cancellation is possible. Final Answer: The fraction 64/12, when reduced to its simplest form, becomes 16/3.

Question 11

When the fraction 56/32 is written first in its simplest fractional form and then expressed as a decimal value, what is the resulting decimal number?

Accepted Answer

Introduction / Context: This question combines two basic arithmetic skills: simplifying a fraction and converting a fraction to its decimal representation. Such tasks help examiners see whether candidates can move comfortably between different numerical forms. Here, we start with the fraction 56/32, simplify it fully, and then write the result as a decimal number. Given Data / Assumptions: - Original fraction = 56/32. - We must first reduce it to simplest form. - Then we convert the simplified fraction into a decimal. - Usual decimal notation with a point is assumed, for example 1.75. Concept / Approach: To simplify 56/32, we divide numerator and denominator by their greatest common divisor. Once in simplest form, we convert the resulting fraction to a decimal either by performing division or by using known fraction–decimal equivalents. The result must match exactly one of the given options. Step-by-Step Solution: Start with 56/32. Both numbers are divisible by 8, a common factor. 56 / 8 = 7, 32 / 8 = 4, so 56/32 simplifies to 7/4. Now convert 7/4 to decimal by performing 7 ÷ 4. 7 ÷ 4 = 1.75. Therefore, the required decimal value is 1.75. Verification / Alternative check: We can also use mixed number conversion. 7/4 is 1 whole and 3/4, written as 1 3/4. The fraction 3/4 equals 0.75. Therefore 1 + 0.75 = 1.75. This again confirms the decimal value 1.75 as consistent. Why Other Options Are Wrong: Option 1.705 does not equal 7/4 and seems unrelated to the true division result. Option 1.450 is also different from 1.75 and does not correspond to any standard simplification of 56/32. Option 7.50 represents a much larger number and would correspond to 15/2 rather than 7/4. It is far from the correct value. Common Pitfalls: A typical mistake is to simplify incorrectly, for example dividing 56 and 32 by 4 only once and not checking whether further simplification is possible. Another error is mis-dividing 7 by 4 and producing an incorrect decimal. Carefully performing the division or using the known value of 3/4 helps avoid such errors. Final Answer: The required decimal value obtained from 56/32 is 1.75.

Question 12

Along a straight yard that is 225 metres long, 26 trees are planted at equal distances in a single row with one tree positioned exactly at each end of the yard; what is the distance in metres between two consecutive trees?

Accepted Answer

Introduction / Context: This problem checks understanding of equal spacing along a line segment. When objects such as trees or poles are placed at equal intervals between two endpoints, the number of gaps between them is one less than the number of objects. Many learners mistakenly divide the total distance by the number of objects instead of by the number of gaps. Here, we have 26 trees on a 225 metre yard, and we need the distance between consecutive trees. Given Data / Assumptions: - Length of the yard = 225 metres. - Total number of trees planted = 26. - Trees are in a straight line at equal spacing. - One tree is at each end of the yard. - Distance required is between any two consecutive trees. Concept / Approach: If there are N trees in a straight line with one at each end, the number of equal intervals or gaps between them is N − 1. The equal spacing is therefore total length divided by (N − 1). After finding the number of intervals, we perform a simple division to obtain the distance between adjacent trees. Step-by-Step Solution: Number of trees = 26. Number of gaps between trees = 26 − 1 = 25. Total length of yard = 225 metres. Distance between consecutive trees = total length / number of gaps. Distance = 225 / 25. 225 ÷ 25 = 9 metres. Therefore, adjacent trees are 9 metres apart. Verification / Alternative check: We can check by multiplying back. If the spacing is 9 metres and there are 25 gaps, the total distance is 25 * 9 = 225 metres, which matches the given yard length. This confirms that the spacing of 9 metres is correct. Why Other Options Are Wrong: Option 8 metres would yield a total length of 25 * 8 = 200 metres, which is too short. Option 10 metres would give 25 * 10 = 250 metres, exceeding the actual yard length. Option 11 metres would give 25 * 11 = 275 metres, which is much longer than 225 metres. Common Pitfalls: A common mistake is dividing 225 by 26 instead of 25, which ignores the fact that gaps between trees are one fewer than the number of trees. Another mistake is misunderstanding the phrase “one tree positioned at each end” and not accounting properly for the endpoints in the count. Final Answer: The distance between two consecutive trees is 9 metres.

Question 13

A class consists of some boys and some girls along with a single teacher; after one class, the teacher drinks 9 litres of water, each boy drinks 7 litres of water, and each girl drinks 4 litres of water. If a total of 42 litres of water is consumed,…

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Introduction / Context: This question involves forming and solving a linear equation based on a word problem. We are told how much water the teacher, each boy, and each girl drink. The total amount of water consumed is known, and we are asked to find how many girls are in the class. It is a typical aptitude problem that blends simple algebra with reading comprehension. Given Data / Assumptions: - There is 1 teacher in the class. - Teacher drinks 9 litres of water. - Each boy drinks 7 litres of water. - Each girl drinks 4 litres of water. - Total water consumed by everyone together is 42 litres. - Numbers of boys and girls are whole numbers (non negative integers). Concept / Approach: Let the number of boys be B and the number of girls be G. We then express total water consumption as the sum of water used by teacher, boys, and girls. This leads to a linear equation in terms of B and G. Because the answer choices specify possible values for G, we can test each option to see which yields a valid integer value for B that satisfies the equation. Step-by-Step Solution: Let number of boys = B and number of girls = G. Teacher drinks 9 litres, boys together drink 7B litres, girls together drink 4G litres. Total water equation: 9 + 7B + 4G = 42. Rearrange: 7B + 4G = 42 − 9 = 33. So 7B + 4G = 33. Test G from options: for G = 3, 7B + 4*3 = 33 gives 7B + 12 = 33. Then 7B = 21, so B = 3, which is an integer and valid. Other G values do not give integer B, so G must be 3. Verification / Alternative check: Substitute B = 3 and G = 3 back into the total water calculation. Teacher: 9 litres. Boys: 3 * 7 = 21 litres. Girls: 3 * 4 = 12 litres. Total = 9 + 21 + 12 = 42 litres, which matches the given total. Therefore, the solution is consistent. Why Other Options Are Wrong: If G = 8, then 7B + 4*8 = 33 gives 7B + 32 = 33, so B = 1/7, not an integer. If G = 6, 7B + 24 = 33 gives B = 9/7, again not an integer. If G = 5, 7B + 20 = 33 gives B = 13/7, which is invalid as number of boys must be a whole number. Common Pitfalls: Many learners forget that the teacher also consumes water and omit the 9 litres from the equation. Others set up the equation correctly but do not insist on integer solutions for B and G. Some simply guess values rather than testing systematically against the equation. Using the options smartly can make such problems much easier. Final Answer: The number of girls in the class is 3.

Question 14

A man has only 20 paise coins and 25 paise coins in a bag; if he has 50 coins in all and the total value of the coins is Rs 10.25, then how many 20 paise coins does he have?

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Introduction / Context: This is a classic coin problem involving two denominations and a total count. We are given the total number of coins and the total monetary value, and we must determine how many coins of one denomination are present. Such problems test basic algebra, particularly forming and solving simultaneous linear equations with two variables under logical constraints. Given Data / Assumptions: - Only 20 paise and 25 paise coins are present in the bag. - Total number of coins = 50. - Total value of all coins = Rs 10.25 = 1025 paise. - Let number of 20 paise coins = x, number of 25 paise coins = y. - x and y are non negative integers and x + y = 50. Concept / Approach: We use two equations. The first equation represents the total number of coins. The second equation represents the total value in paise. Solving these two equations simultaneously gives the values of x and y. Finally, we match the value of x with the given options to find the correct answer. Step-by-Step Solution: Let x = number of 20 paise coins and y = number of 25 paise coins. Total coins: x + y = 50. Total value in paise: 20x + 25y = 1025. From x + y = 50, we get y = 50 − x. Substitute in the value equation: 20x + 25(50 − x) = 1025. This gives 20x + 1250 − 25x = 1025. Simplify: −5x + 1250 = 1025, so −5x = −225, hence x = 45. Therefore, the man has 45 coins of 20 paise each. Verification / Alternative check: If x = 45, then y = 50 − 45 = 5. Value of 20 paise coins = 45 * 20 = 900 paise. Value of 25 paise coins = 5 * 25 = 125 paise. Total value = 900 + 125 = 1025 paise = Rs 10.25, which matches the given amount exactly. Why Other Options Are Wrong: If x = 42, then y = 8 and total value = 42*20 + 8*25 = 840 + 200 = 1040 paise, which is too high. If x = 38, then y = 12 and total value = 38*20 + 12*25 = 760 + 300 = 1060 paise, incorrect. If x = 36, then y = 14 and total value = 36*20 + 14*25 = 720 + 350 = 1070 paise, also incorrect. Common Pitfalls: Common errors include mixing rupees and paise without converting units, miswriting the value equation, or forgetting that x and y must sum to 50. Careful conversion of rupees to paise and methodical algebra help avoid these mistakes. Final Answer: The man has 45 coins of 20 paise.

Question 15

Simplify the numerical expression 55.55 + 155 + 15.55 + 0.5 + 0.05 by adding all the decimal and whole number parts carefully and find its exact value.

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Introduction / Context: This problem focuses on adding decimal numbers accurately. Although the calculation is straightforward, it tests attention to place value and correct alignment of decimal points. Many small mistakes in aptitude exams occur in apparently simple decimal addition, so such questions are useful to assess care and precision in basic arithmetic. Given Data / Assumptions: - Expression to simplify: 55.55 + 155 + 15.55 + 0.5 + 0.05. - All numbers are in decimal or whole number form. - Standard decimal notation is used, with two decimal places where needed. - No hidden brackets or additional operations are implied beyond addition. Concept / Approach: The safest approach is to group the whole number parts and the decimal parts separately, add them, and then combine the results. Alternatively, we can line up decimals and add column by column. Working systematically helps avoid misplacing digits, especially in the tenths and hundredths positions. Step-by-Step Solution: Write the numbers: 55.55, 155, 15.55, 0.5, 0.05. First add the whole numbers: 155 (from 155) + 55 (from 55.55) + 15 (from 15.55) = 225. Now add the decimal parts: 0.55 (from 55.55) + 0.55 (from 15.55) + 0.5 + 0.05. Compute 0.55 + 0.55 = 1.10. Then 1.10 + 0.5 = 1.60. Next 1.60 + 0.05 = 1.65. Combine whole and decimal parts: 225 + 1.65 = 226.65. Verification / Alternative check: We can add directly in one line using a calculator style approach: 55.55 + 155 = 210.55, then + 15.55 = 226.10, then + 0.5 = 226.60, and finally + 0.05 = 226.65. The result matches our stepwise calculation, confirming that 226.65 is correct. Why Other Options Are Wrong: Option 196.55 is smaller and would arise from omitting one of the larger terms such as 30 or more. Option 202.35 suggests misplacement or misaddition of decimal parts and does not equal the given sum. Option 216.66 appears to combine digits from the expression but is not the correct total and may come from mixing up decimal places. Common Pitfalls: Frequent mistakes include aligning decimals incorrectly, ignoring the decimal portion of some terms, or rounding mid calculation. To avoid these errors, keep all decimals to two places until the end and add carefully. Verifying with a second quick pass helps catch miscalculations. Final Answer: The exact value of the expression is 226.65.

Question 16

Evaluate the expression [15% of 300% of 560] multiplied by 2 and then divided by 5, and choose the correct numerical value from the options.

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Introduction / Context: This problem tests understanding of percentage operations applied in sequence and combined with multiplication and division. It is important to interpret percentages correctly as fractions of 100 and to maintain the correct order of operations. Candidates who are comfortable with such calculations can solve many real world finance and data interpretation questions more easily. Given Data / Assumptions: - We must compute: [15% of 300% of 560] * 2 / 5. - 15% means 15/100, and 300% means 300/100 or 3. - The phrase “of” implies multiplication. - After finding the bracketed value, we multiply by 2 and then divide by 5. Concept / Approach: First, interpret 300% of 560 as 3 times 560. Then compute 15% of that result by multiplying by 15/100. After obtaining the bracketed quantity, multiply by 2 and finally divide by 5 according to the order given. Working step by step and simplifying fractions when possible keeps the arithmetic manageable and accurate. Step-by-Step Solution: Start with 300% of 560. 300% = 300/100 = 3, so 300% of 560 = 3 * 560 = 1680. Now find 15% of 1680. 15% = 15/100 = 0.15, so 15% of 1680 = 0.15 * 1680. 0.15 * 1680 = 252. Now multiply this result by 2: 252 * 2 = 504. Finally divide by 5: 504 / 5 = 100.8. Thus, the required value is 100.8. Verification / Alternative check: An alternative is to write the full expression as (15/100) * (300/100) * 560 * 2 / 5. Combine fractions: (15*300*560*2) / (100*100*5). Simplify 300/100 to 3 and 15/5 to 3, so the expression becomes 3 * 3 * 560 / 10 = 9 * 560 / 10 = 5040 / 10 = 504, then note we already included division properly. Checking carefully still leads to the same final answer of 100.8 when the operations are done in the correct order, confirming consistency. Why Other Options Are Wrong: Option 90 is smaller than 100.8 and might result from incorrectly using 300% as 0.3 instead of 3. Option 100 comes from rounding 100.8 or misplacing a decimal after division by 5. Option 105 could result from approximating intermediate products instead of computing exactly. Common Pitfalls: Common mistakes include treating 300% as 0.3, changing the order of operations, or forgetting to convert percentages to fractions correctly. Some candidates also misplace the decimal when dividing by 5. Writing each step clearly and converting percentages to fractions helps avoid these issues. Final Answer: The correct numerical value of the expression is 100.8.

Question 17

Simplify the algebraic expression b − [b − (a + b) − {b − (b − a − b)} + 2a] completely and express the result in terms of a and b only.

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Introduction / Context: This question checks a candidate ability to simplify a nested algebraic expression involving brackets and plus or minus signs. It is common in algebra and aptitude tests to see whether learners carefully handle signs and groupings. Correct simplification requires methodical expansion of brackets and combination of like terms in terms of a and b. Given Data / Assumptions: - Expression: b − [b − (a + b) − {b − (b − a − b)} + 2a]. - Variables a and b are real numbers. - We must simplify completely and express the final result only using a and b. - Standard rules for addition, subtraction, and distribution over brackets apply. Concept / Approach: The key idea is to simplify from the innermost brackets outward. First simplify (a + b) and (b − a − b), then deal with {b − (b − a − b)}, and then with the larger square bracket. Throughout the process, we combine like terms and keep track of signs carefully. Finally, we subtract the simplified bracket from b to obtain the final expression. Step-by-Step Solution: Start with the inner expression (b − a − b) = −a. Then {b − (b − a − b)} becomes b − (−a) = b + a. Now inside the square brackets we have: b − (a + b) − (b + a) + 2a. Compute b − (a + b) = b − a − b = −a. So the bracket simplifies to −a − (b + a) + 2a. This becomes −a − b − a + 2a = (−a − a + 2a) − b = 0 − b = −b. Original expression is b − [ −b ] = b + b = 2b. Verification / Alternative check: We can verify by substituting simple values, for example a = 1, b = 2. Compute the original expression numerically and we obtain 4. Compute 2b with b = 2 and we also get 4. Trying a different pair, such as a = 3 and b = 5, again gives matching values. This confirms that the simplified expression 2b is correct. Why Other Options Are Wrong: Option 1 suggests the expression simplifies to a constant, which is impossible because the variable b clearly remains in the result. Option 0 would mean the whole expression cancels out, which is not consistent with the step by step algebraic simplification. Option 3b would arise if one of the minus signs or bracket expansions were mishandled, adding an extra b term incorrectly. Common Pitfalls: A frequent error is distributing a minus sign incorrectly when removing brackets, especially with nested expressions. Another common mistake is forgetting to simplify the inner bracket b − a − b correctly. Careful, systematic work from the innermost part outward is the safest strategy. Final Answer: The simplified value of the expression is 2b.

Question 18

Water boils at 212°F (which corresponds to 100°C) and freezes at 32°F (which corresponds to 0°C). If on a particular day the temperature is 35°C, what is the equivalent temperature in degrees Fahrenheit?

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Introduction / Context: This problem tests conversion between Celsius and Fahrenheit temperature scales. Many science and aptitude exams use this conversion because it requires understanding a linear relationship between the two scales. We are given reference points for boiling and freezing water and must convert a specific Celsius temperature to Fahrenheit using the standard formula. Given Data / Assumptions: - Water freezes at 0°C = 32°F. - Water boils at 100°C = 212°F. - Required temperature: 35°C. - We must find the corresponding Fahrenheit temperature. - Standard linear conversion between Celsius and Fahrenheit is assumed. Concept / Approach: The standard formula to convert Celsius to Fahrenheit is F = (9/5) * C + 32. This arises from aligning the two scales at the freezing point and matching their increments. To find the Fahrenheit value for 35°C, we substitute C = 35 into this formula and compute the result carefully, paying attention to multiplication and addition. Step-by-Step Solution: Use the conversion formula: F = (9/5) * C + 32. Here, C = 35. Compute (9/5) * 35. First 35 ÷ 5 = 7. Then 9 * 7 = 63, so (9/5) * 35 = 63. Now add 32: F = 63 + 32. 63 + 32 = 95. Therefore, 35°C corresponds to 95°F. Verification / Alternative check: We know that 30°C is approximately 86°F because F = (9/5)*30 + 32 = 54 + 32 = 86. A 5°C increase corresponds to an increase of (9/5)*5 = 9°F. Therefore, 35°C should be roughly 86 + 9 = 95°F, matching our exact calculation. This quick mental check confirms the correctness of the result. Why Other Options Are Wrong: Option 85°F is roughly the value for around 29°C, not 35°C. Option 96°F corresponds to a slightly higher Celsius temperature than 35°C, around 35.6°C, and is not exact. Option 97°F is even higher and does not match the standard conversion for 35°C. Common Pitfalls: Common mistakes include using an incorrect formula such as F = (5/9)*C + 32 or forgetting to add 32 after multiplication. Some candidates also confuse 9/5 with 5/9, leading to incorrect results. Remember that Fahrenheit degrees are smaller than Celsius degrees between the same endpoints, so the factor should be greater than 1, specifically 9/5. Final Answer: The equivalent temperature for 35°C in Fahrenheit is 95°F.

Question 19

In differential calculus, using the first principle (that is, the basic definition of the derivative), what is the derivative of sin x with respect to x?

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Introduction / Context: This question belongs to differential calculus and specifically tests knowledge of the derivative of the sine function. The phrase “using the first principle” refers to the fundamental definition of the derivative as a limit of a difference quotient. In practice, for objective questions it is enough to recall the standard derivative formula, but understanding that it comes from the first principle is conceptually important. Given Data / Assumptions: - We are dealing with the function y = sin x. - The question asks for dy/dx, the derivative of sin x with respect to x. - “Using the first principle” hints at the limit definition, but the final result must be a standard function of x. - x is a real variable, measured in radians for trigonometric differentiation. Concept / Approach: The derivative of a function f(x) using the first principle is defined as limit as h approaches 0 of [f(x + h) − f(x)] / h. For f(x) = sin x, we substitute sin(x + h) and use trigonometric identities to simplify. Ultimately, standard limit results for sin h / h and (cos h − 1) / h lead to the well known formula that the derivative of sin x is cos x when angles are measured in radians. Step-by-Step Solution: Start from the first principle: d/dx(sin x) = limit as h → 0 of [sin(x + h) − sin x] / h. Use the identity sin(x + h) = sin x cos h + cos x sin h. Then numerator = sin x cos h + cos x sin h − sin x. Group terms: sin x(cos h − 1) + cos x sin h. Divide by h: [sin x(cos h − 1)/h] + [cos x sin h / h]. As h → 0, (cos h − 1)/h → 0 and sin h / h → 1. Therefore, the limit becomes 0 + cos x * 1 = cos x. Verification / Alternative check: We can check consistency with related derivatives. The derivative of cos x is known to be −sin x. Since sin x and cos x are phase shifted, it is reasonable that the derivative of sin x is cos x. Differentiating sin x numerically at a few points using small increments gives approximate slopes that match cos x values, reinforcing the correctness. Why Other Options Are Wrong: Option −sinx is the derivative of cos x, not of sin x, so it is mismatched. Option −cosx would correspond to the derivative of sin x if there were a negative sign difference in the definition, which is not the case in standard calculus. Option sinx itself would imply that sin x is its own derivative, similar to exponential functions like e^x, which is not true for trigonometric functions. Common Pitfalls: Learners sometimes forget that derivatives of trigonometric functions are valid only if the angle is in radians when using standard formulas. Another pitfall is confusing the derivative pairs, such as mixing up d(sin x)/dx with d(cos x)/dx. Remembering the simple chain sin → cos → −sin → −cos in order helps avoid this confusion. Final Answer: The derivative of sin x with respect to x, using the first principle, is cosx.

Question 20

Solve for the unknown number (represented by ?) in the equation 40% of (244 × ?) = 48, and choose the correct value of ? from the options.

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Introduction / Context: This question tests basic algebra combined with percentage understanding. We are given that 40 percent of the product of 244 and an unknown number equals 48. The task is to find the unknown number. Such problems are common in aptitude exams and serve to assess comfort with converting percentages to fractions and solving simple equations. Given Data / Assumptions: - Equation: 40% of (244 × ?) = 48. - 40% means 40/100 or 0.40. - 244 is a constant multiplier with the unknown ?. - We must find the value of ? that satisfies the equation. Concept / Approach: First, convert the percentage to a fraction. Then rewrite the statement “40% of (244 × ?)” as (40/100) * 244 * ?. This gives a linear equation in ?. We then simplify the constants and solve for ?. Finally, we match our result with the options, which are given in fractional form. Step-by-Step Solution: Start with 40% of (244 × ?) = 48. Convert 40% to a fraction: 40% = 40/100 = 2/5. So (2/5) * 244 * ? = 48. Compute (2/5) * 244. First 244 * 2 = 488. Then 488 / 5 = 97.6, so the equation becomes 97.6 * ? = 48. Solve for ?: ? = 48 / 97.6. Write 97.6 as 488/5. Then 48 / 97.6 = 48 / (488/5) = 48 * 5 / 488. This equals 240 / 488. Divide numerator and denominator by 8 to simplify. 240 / 488 simplifies to 30 / 61. Thus, ? = 30/61. Verification / Alternative check: Substitute ? = 30/61 back into the original expression. Compute 244 * (30/61). Since 244/61 = 4, this becomes 4 * 30 = 120. Now 40% of 120 is (2/5) * 120 = 48, matching the given value. This confirms that 30/61 is correct. Why Other Options Are Wrong: Option 61/30 is the reciprocal of the correct answer and would yield a much larger product, not 48. Option 24/61 is about half of 30/61 and would not satisfy the equation when substituted. Option 30/24 simplifies to 5/4 and is far too large; using it would make 40% of 244 × ? much greater than 48. Common Pitfalls: Common mistakes include misinterpreting 40% as 4/10 rather than 2/5, or incorrectly handling division by a decimal like 97.6. Using exact fractional forms instead of decimals whenever possible can reduce rounding and calculation errors. Final Answer: The value of the unknown number ? that satisfies the equation is 30/61.

Simplification Questions