Question 1

The recurring decimal 0.125125125... has the block 125 repeating without end. What is the rational number (fraction in simplest form) that is exactly equal to this decimal?

Accepted Answer

Introduction / Context: This problem again explores the connection between recurring decimals and rational numbers. Here, the repeating block has three digits, so you must be comfortable with the algebraic method for handling longer repeating patterns and converting them into fractions in lowest terms. Given Data / Assumptions: The decimal is 0.125125125..., where the block 125 repeats indefinitely. The repetition is continuous with no extra digits breaking the pattern. We must find the exact fraction that represents this number. The final answer should be in simplest form. Concept / Approach: The method is similar to the earlier recurring decimal question but now uses a power of 10 that matches the length of the repeating block, which is three digits. We set the decimal equal to a variable, multiply by 1000 to shift one full block to the left of the decimal point, subtract the original variable, and solve the resulting linear equation. This reliably eliminates the repeating part and yields a clean fraction. Step-by-Step Solution: Let x = 0.125125125..., with 125 repeating.Because the repeating block length is three digits, multiply both sides by 1000: 1000x = 125.125125...Now subtract the original x: 1000x - x = 125.125125... - 0.125125...This simplifies to 999x = 125.Solve for x: x = 125 / 999.Check if 125 / 999 is in simplest form. The prime factors of 125 are 5 × 5 × 5, while 999 factors into 3 × 3 × 3 × 37. There is no common factor other than 1.Therefore, 0.125125125... = 125 / 999 in simplest form. Verification / Alternative check: To verify, perform long division of 125 by 999. The quotient will be 0.125125125..., with the block 125 repeating indefinitely. Any scientific calculator that can show repeating decimals or enough digits will support this, as the digits will follow the pattern 125 without deviation. This confirms that our algebraic procedure and simplification are correct. Why Other Options Are Wrong: Option 125/99 corresponds to a repeating block of two digits, not three, and its decimal expansion does not show the pattern 125. Option 125/9999 involves four nines in the denominator and would produce a different repeating cycle. Option 125/9 gives a very large repeating decimal with a different pattern. Option 25/198 is a reduced fraction with a completely different decimal representation. None of these match the exact pattern of 0.125125125.... Common Pitfalls: Some learners mistakenly multiply by 10 or 100 instead of 1000, which fails to align the repeating blocks correctly. Others forget that the denominator must be 10^n - 1 for a repeating block of length n, and thus they guess denominators without algebraic justification. Forgetting to check whether the fraction can be simplified is another common oversight. Careful factorization ensures that the final answer is in lowest terms. Final Answer: The recurring decimal 0.125125125... is exactly equal to the fraction 125/999 in simplest form.

Question 2

When 4.036 is divided by 0.04, what is the resulting quotient?

Accepted Answer

Introduction / Context: This question focuses on division of decimals and the effect of moving decimal points in both the dividend and divisor. It is a basic but important skill for handling decimal operations without always relying on a calculator and appears frequently in decimal fraction sections of aptitude exams. Given Data / Assumptions: Dividend (number being divided) = 4.036. Divisor (number we divide by) = 0.04. We are asked to compute the quotient 4.036 ÷ 0.04. All values are exact decimals, and no rounding is requested. Concept / Approach: To divide by a decimal, a standard approach is to convert the divisor into a whole number by multiplying both the dividend and divisor by the same power of 10. This keeps the value of the quotient unchanged but allows us to perform a simpler division by a whole number. Specifically, if the divisor has two digits after the decimal, we multiply both numbers by 100 to remove the decimal from the divisor. Step-by-Step Solution: We need to evaluate 4.036 ÷ 0.04.The divisor 0.04 has two decimal places. Multiply both numbers by 100.4.036 × 100 = 403.6.0.04 × 100 = 4.Now compute the simpler quotient: 403.6 ÷ 4.Perform the division: 4 goes into 40 ten times, leaving 0; bring down 3 to get 3.0, which gives 0 with remainder 3. Bring down 6 to get 36, which gives 9.Thus 403.6 ÷ 4 = 100.9.Therefore, 4.036 ÷ 0.04 = 100.9. Verification / Alternative check: We can check the result by reversing the operation. Multiply the quotient by the divisor: 100.9 × 0.04. First, 100.9 × 4 = 403.6. Then move the decimal two places left to account for dividing by 100, giving 4.036. This matches the original dividend exactly, so our quotient 100.9 is confirmed as correct. Why Other Options Are Wrong: Option 1.009 would correspond to dividing 4.036 by 4, not by 0.04. Option 10.09 is off by a factor of 10 and results from shifting the decimal incorrectly only once. Option 1009 is a factor of 10 larger than the correct answer and would imply an incorrect scaling. Option 0.1009 is far too small and does not reflect division by a number less than 1, which should increase the result, not decrease it. Common Pitfalls: Common errors include moving the decimal point in only the divisor but not the dividend, or shifting the decimal in opposite directions. Another frequent mistake is forgetting that dividing by a small decimal (less than 1) should produce a larger quotient. Keeping track of how many decimal places you shift and always adjusting both numbers by the same power of 10 help to avoid these mistakes. Final Answer: The quotient when 4.036 is divided by 0.04 is 100.9.

Question 3

Karthik read 6/13 of a book in the first week and 5/9 of the remaining pages in the second week. If 100 pages were still unread after the second week, approximately how many pages were there in the entire book?

Accepted Answer

Introduction / Context: This question combines fraction operations with a word problem involving reading progress. It tests your ability to interpret fractional parts of a whole, handle successive fractions on a remaining quantity, and then work backward from the remaining pages to estimate the total number of pages in the book. Given Data / Assumptions: Let the total number of pages in the book be N. In the first week, Karthik reads 6/13 of the book. In the second week, he reads 5/9 of the remaining pages. After the second week, 100 pages are still unread. We assume the fractions are applied exactly, and the final answer is chosen from the nearest integer value among the options. Concept / Approach: The key idea is to track what fraction of the book remains after each week. First, subtract the fraction read in the first week to find the remaining part. Then apply the second fraction to that remainder to find how much more is read. The remaining unread portion after two weeks is set equal to 100 pages. This leads to an equation in N, which we solve and then compare to the options, choosing the nearest whole number of pages. Step-by-Step Solution: Let total pages be N.Pages read in first week = (6/13)N, so remaining after week one = N - (6/13)N = (7/13)N.In the second week, Karthik reads 5/9 of the remaining pages: (5/9) × (7/13)N = (35/117)N.Remaining after second week = (7/13)N - (35/117)N.Convert 7/13 to a denominator of 117: 7/13 = 63/117.So remaining fraction = (63/117 - 35/117)N = (28/117)N.We are told this remainder equals 100 pages, so (28/117)N = 100.Solve for N: N = 100 × (117/28) ≈ 417.857.The closest whole number among the options is 418 pages. Verification / Alternative check: Take N = 418 and recompute the process. After week one, remaining pages = (7/13) × 418 ≈ 225.077, which we interpret as about 225 pages. Five ninths of this is approximately 125 pages, leaving close to 100 pages unread. Because the problem is drawn from an aptitude style context and the options are discrete integers, 418 is the best match, and small rounding adjustments are acceptable in this type of exam question. Why Other Options Are Wrong: Taking N as 404, 415, or 420 and repeating the same fraction operations does not produce a remaining count close to exactly 100 pages. The deviations are larger than those obtained with 418, so they are poorer approximations. Option 390 is clearly too small and leads to a remaining count significantly different from 100 pages. Among all choices, 418 fits the derived equation most closely. Common Pitfalls: Students often misinterpret the second fraction as 5/9 of the entire book rather than 5/9 of the remaining pages. Another error is adding or subtracting fractions with different denominators incorrectly, which leads to the wrong remaining fraction. Some learners also forget to equate the remaining fraction of the book to 100 pages and instead subtract 100 at an incorrect stage. Careful stepwise tracking of the fractions at each stage avoids these mistakes. Final Answer: The total number of pages in the book is best taken as 418 pages based on the given data and available options.

Question 4

If two fractions each have values strictly between 0 and 1, what can be said about the value of their product?

Accepted Answer

Introduction / Context: This conceptual question examines how multiplication behaves for fractions between 0 and 1. Understanding this property is fundamental in number theory and probability, where such fractions often represent probabilities, proportions, or scaling factors that shrink quantities rather than enlarge them. Given Data / Assumptions: We are dealing with two fractions, call them a and b. Both fractions satisfy 0 < a < 1 and 0 < b < 1. We must determine how the product a × b compares to a and b individually. The options describe different possible relationships between the product and the original fractions. Concept / Approach: When you multiply a positive number by a fraction less than 1, the result becomes smaller than the original number. This is because a fraction less than 1 represents a part of the whole. If you apply such a reduction twice, the final result must be smaller than both of the original positive fractions. Thus, the product of two numbers between 0 and 1 should always be less than each of them. Step-by-Step Solution: Let 0 < a < 1 and 0 < b < 1.First, consider a × b compared to a. Since 0 < b < 1, multiplying a by b gives a × b < a.Next, consider a × b compared to b. Since 0 < a < 1, multiplying b by a gives a × b < b.Therefore, a × b is less than both a and b.Because both a and b are positive, the product is still positive, but it is strictly smaller than each factor.Hence, the product is always less than either of the original fractions. Verification / Alternative check: We can verify this with a simple numerical example. Take a = 1/2 and b = 3/4. Both are between 0 and 1. Their product is (1/2) × (3/4) = 3/8 = 0.375. Compare: 0.375 < 0.5 and 0.375 < 0.75. This confirms the general reasoning. Trying other pairs of fractions such as 0.2 and 0.6 gives 0.12, which again is smaller than both 0.2 and 0.6. Why Other Options Are Wrong: Option stating that the product is always greater than either fraction contradicts the basic shrinking effect of multiplying by a number less than 1. The option suggesting that the product can sometimes be greater is also incorrect given the strict inequalities. Saying it remains the same or equals one original fraction would require one factor to be 1, which is excluded because both fractions are strictly less than 1. The product certainly does not equal the sum, which would be larger than either fraction. Common Pitfalls: A common misunderstanding is to think of multiplication as always making numbers larger, based on experience with whole numbers. This intuition fails for fractions less than 1. Another mistake is confusing multiplication with addition, mistakenly expecting the result to lie between the two fractions as a kind of average. Remember that multiplying by a fraction less than 1 always scales the quantity down, not up. Final Answer: The product of two fractions each between 0 and 1 is always less than either of the original fractions.

Question 5

How many digits will appear to the right of the decimal point in the product of 95.75 and 0.2554, based on decimal place counting rules?

Accepted Answer

Introduction / Context: This question is about understanding how many decimal places appear in the product of two decimal numbers. It targets a very common rule taught in schools: the total number of digits to the right of the decimal point in a product equals the sum of the digits to the right of the decimal point in the factors, ignoring later simplification or cancellation. Given Data / Assumptions: We are multiplying 95.75 by 0.2554. 95.75 has two digits after the decimal point. 0.2554 has four digits after the decimal point. We are asked how many digits will be to the right of the decimal point in the product by using the standard counting rule. Concept / Approach: The usual method for estimating the number of decimal digits in a product is to ignore the decimal points, multiply the numbers as integers, and then count the total number of decimal places by adding the decimal digits from each factor. That sum tells us how many digits to place to the right of the decimal in the final result before any simplification is considered. Step-by-Step Solution: First count the decimal places in 95.75. It has two digits after the decimal: 7 and 5.Next count the decimal places in 0.2554. It has four digits after the decimal: 2, 5, 5, and 4.Add these together: 2 + 4 = 6.Therefore, when 95.75 and 0.2554 are multiplied, the product will have six digits to the right of the decimal point according to the standard rule.So the answer is 6. Verification / Alternative check: If you actually perform the multiplication or use a calculator, you will find that the product is 24.45455. In this specific case, the full product may simplify and appear with fewer effective decimal digits. However, the method taught in aptitude and basic arithmetic typically asks you to apply the counting rule based on the original decimal places before simplification, which gives six decimal places as the expected result. The question is clearly focused on this rule rather than on simplification details. Why Other Options Are Wrong: Options 7, 8, and 9 assume more decimal digits than the sum of the original decimal places, which contradicts the rule. Option 5 assumes fewer digits and arises from noticing that the actual decimal representation may simplify, but that is not what the standard classroom rule asks you to count. The teaching focus is on adding the decimal places from both factors, leading to exactly six digits to the right of the decimal point. Common Pitfalls: Some learners mistakenly count the digits to the left of the decimal or mix them up with digits to the right. Others believe that the number of decimal places might change unpredictably with multiplication, which is not true when you consider the raw product before simplification. Always count the digits to the right of the decimal in each factor and add them to get the total decimal places in the product. Final Answer: Based on the standard decimal counting rule, the product of 95.75 and 0.2554 will have 6 digits to the right of the decimal point.

Question 6

Evaluate the product 0.002 × 0.5 and express the result as a decimal.

Accepted Answer

Introduction / Context: This is a basic decimal multiplication question and tests your understanding of how multiplying by a fraction less than 1 affects the value of a small decimal. Such problems appear in the fundamentals of decimal fractions, percentages, and measurement conversions. Given Data / Assumptions: We need to compute 0.002 × 0.5. Both numbers are positive decimals. We want the exact decimal value of the product. No rounding is required, and all options are simple decimals. Concept / Approach: We recall that 0.5 is equal to 1/2. Multiplying a number by 1/2 is the same as taking half of that number. So this problem can be interpreted as finding half of 0.002. Alternatively, we can treat both numbers as decimals and multiply directly, keeping track of the total number of decimal places. Step-by-Step Solution: Interpret 0.5 as 1/2.Compute half of 0.002: 0.002 ÷ 2 = 0.001.Therefore, 0.002 × 0.5 = 0.001.We can also confirm by counting decimal places using direct multiplication.Write 0.002 as 2 × 10^-3 and 0.5 as 5 × 10^-1.Multiply: (2 × 10^-3) × (5 × 10^-1) = 10 × 10^-4 = 10^-3 = 0.001.Both methods lead to the same result. Verification / Alternative check: Another way is to convert to fractions. 0.002 = 2/1000 and 0.5 = 1/2. Multiply: (2/1000) × (1/2) = 2 / 2000 = 1 / 1000. The fraction 1 / 1000 corresponds to the decimal 0.001. This fraction based method confirms the decimal calculation and shows the consistency of the answer across different approaches. Why Other Options Are Wrong: Option 0.0001 is ten times smaller than the correct value and would correspond to mistakenly shifting the decimal one extra place to the left. Option 0.01 and 0.1 are significantly larger and reflect a misunderstanding that multiplying by 0.5 should make the number larger, which is not true. Option 0.00001 is even smaller and corresponds to two extra left shifts. Only 0.001 correctly represents half of 0.002. Common Pitfalls: Some students confuse multiplication by 0.5 with division by 0.5, which would double the number instead of halving it. Another mistake is miscounting decimal places when multiplying small decimals, often introducing one or more extra zeros. A useful habit is to interpret 0.5 as 1/2 and think in terms of taking half, which provides a strong intuitive check on the size of the final answer. Final Answer: The product 0.002 × 0.5 is equal to 0.001.

Question 7

The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. In 2001, the price of X was Rs. 4.20 and the price of Y was Rs. 6.30. In which year will commodity X cost exactly 40 paise mo…

Accepted Answer

Introduction / Context: This question involves linear growth of prices over time. It tests your ability to model year on year increments as simple arithmetic progressions and then solve for the time when a given difference between two prices is achieved. Such questions are common in topics related to simple interest, progressions, and basic algebra. Given Data / Assumptions: In 2001, price of commodity X = Rs. 4.20. In 2001, price of commodity Y = Rs. 6.30. Price of X increases by 40 paise (Rs. 0.40) every year. Price of Y increases by 15 paise (Rs. 0.15) every year. We must find the year when price of X is exactly 40 paise more than price of Y. Concept / Approach: We treat the price of each commodity as a linear function of the number of years after 2001. Let n be the number of years after 2001. Then we express the prices of X and Y in terms of n using their annual increases. We then impose the condition that X is 0.40 rupee more than Y and solve the resulting linear equation for n, finally converting n into the actual calendar year. Step-by-Step Solution: Let n be the number of years after 2001.Price of X after n years: X(n) = 4.20 + 0.40n.Price of Y after n years: Y(n) = 6.30 + 0.15n.We want X(n) to be 0.40 higher than Y(n): X(n) = Y(n) + 0.40.Substitute the expressions: 4.20 + 0.40n = 6.30 + 0.15n + 0.40.Simplify the right side: 6.30 + 0.40 = 6.70, so 4.20 + 0.40n = 6.70 + 0.15n.Rearrange: 0.40n - 0.15n = 6.70 - 4.20.This gives 0.25n = 2.50.Solve for n: n = 2.50 / 0.25 = 10.The required year is 2001 + 10 = 2011. Verification / Alternative check: Compute prices in 2011 directly. After 10 years, price of X = 4.20 + 10 × 0.40 = 4.20 + 4.00 = Rs. 8.20. Price of Y in 2011 = 6.30 + 10 × 0.15 = 6.30 + 1.50 = Rs. 7.80. The difference is 8.20 - 7.80 = Rs. 0.40, which matches the requirement that X be 40 paise more than Y. This confirms that 2011 is the correct year. Why Other Options Are Wrong: If you try 2010 (n = 9), prices become X = 4.20 + 3.60 = 7.80 and Y = 6.30 + 1.35 = 7.65, difference = 0.15. For 2012 (n = 11), the difference is 0.65. For 2013, the difference becomes even larger. None of these match the required 0.40 difference. Therefore, only the year 2011 fits the condition exactly. Common Pitfalls: One common mistake is to set X equal to Y instead of X equal to Y plus 0.40. Another error is mixing rupees and paise improperly, for example by working in paise for one quantity and rupees for another. Some students might also misinterpret 2001 as year zero and then add or subtract incorrectly. Keeping all amounts in rupees and defining n clearly as the number of years after 2001 helps maintain accuracy. Final Answer: Commodity X will cost exactly 40 paise more than commodity Y in the year 2011.

Question 8

What is the decimal quotient obtained when 18 is divided by 1000?

Accepted Answer

Introduction / Context: This question tests simple decimal division by a power of 10. Dividing by 10, 100, 1000, and so on is a fundamental skill that allows quick mental calculations without long division, since it corresponds to shifting the decimal point to the left. Given Data / Assumptions: We need to compute 18 ÷ 1000. Both numbers are positive. The divisor 1000 is a power of 10. The answer must be expressed as a decimal. Concept / Approach: Dividing by 10, 100, or 1000 shifts the decimal point of the dividend to the left by 1, 2, or 3 places respectively. The number 18 can be written as 18.0. Since 1000 = 10^3, dividing 18 by 1000 requires shifting the decimal point three places to the left, filling in zeros where needed. Step-by-Step Solution: Write 18 as 18.0 to visualize the decimal point.Because 1000 = 10^3, we need to move the decimal point three places to the left.Move one place: 1.80.Move two places: 0.180.Move three places: 0.0180.So 18 ÷ 1000 = 0.018.Thus, the decimal quotient is 0.018. Verification / Alternative check: We can check with fractions. Write 18 ÷ 1000 as 18/1000. Both numerator and denominator are divisible by 2, giving 9/500. To convert 9/500 to a decimal, note that 500 = 5 × 100, so 9/500 = (9/5) ÷ 100 = 1.8 ÷ 100 = 0.018. This confirms that the decimal representation is correct and consistent with the place value method. Why Other Options Are Wrong: Option 1.8 corresponds to dividing by 10, not 1000. Option 0.18 corresponds to dividing by 100. Option 0.0018 would be obtained by dividing by 10,000. Option 0.00018 reflects an even larger divisor. All of these are incorrect for division by 1000. Only 0.018 correctly represents the quotient when 18 is divided by 1000. Common Pitfalls: Students sometimes move the decimal point the wrong way, to the right instead of the left, which effectively multiplies instead of dividing. Another mistake is to move the decimal the wrong number of places, for example shifting only twice for division by 1000. Carefully counting the number of zeros in the divisor and moving the decimal left by that same number is the key to avoiding these errors. Final Answer: The decimal quotient of 18 divided by 1000 is 0.018.

Question 9

For which value of x will the expression 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02 represent a perfect square?

Accepted Answer

Introduction / Context: This problem tests your recognition of algebraic identities in decimal form, specifically the expansion of a perfect square. Recognizing patterns like (a + b)^2 and matching them to given expressions is a powerful technique for simplifying algebra and for quickly identifying missing coefficients. Given Data / Assumptions: The expression is 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02. We are told that for some value of x this expression becomes a perfect square. The options provide possible values for x: 0.01, 0.02, 0.03, 0.04, and 0.05. We must determine the value of x that makes the entire expression equal to (11.98 + 0.02)^2 or a similar pattern. Concept / Approach: Recall the identity (a + b)^2 = a^2 + 2ab + b^2. In the given expression, 11.98 × 11.98 plays the role of a^2, 0.02 × 0.02 plays the role of b^2, and 11.98 × x should correspond to the middle term 2ab. We match these pieces and solve for x by equating 11.98 × x with 2 × 11.98 × 0.02. Step-by-Step Solution: Identify a and b: a = 11.98 and b = 0.02.A perfect square of the form (a + b)^2 expands to a^2 + 2ab + b^2.In our expression, a^2 = 11.98 × 11.98 and b^2 = 0.02 × 0.02, so those parts already match.The middle term in the perfect square would be 2ab = 2 × 11.98 × 0.02.Compute 2 × 11.98 × 0.02 = 11.98 × 0.04.In the given expression, the middle term is 11.98 × x.For the expressions to match, we must have 11.98 × x = 11.98 × 0.04.Therefore, x = 0.04. Verification / Alternative check: Substitute x = 0.04 back into the expression. It becomes 11.98 × 11.98 + 11.98 × 0.04 + 0.02 × 0.02. This matches the expansion of (11.98 + 0.02)^2 exactly, which is 12.00^2. Therefore, the expression becomes 144, a perfect square. No other value of x will produce this exact algebraic identity, so 0.04 is the unique correct answer among the choices. Why Other Options Are Wrong: Any other value for x would make the middle term 11.98 × x different from the required 2ab term. For example, x = 0.02 would give 11.98 × 0.02, which is only half the required value. Similarly, x = 0.03 or 0.05 would give mismatched coefficients, meaning the expression would no longer be a perfect square and would not simplify to (a + b)^2 for any a and b. Common Pitfalls: One common mistake is to confuse the structure of the identity and think that the middle term should be a × b instead of 2ab. This leads to an incorrect factor of 2 in the value of x. Another error is to try plugging in each option directly without recognizing the identity, which can be time consuming and prone to arithmetic slips. Spotting the pattern a^2 + 2ab + b^2 and using it directly is the fastest and most reliable method. Final Answer: The expression becomes a perfect square when x is 0.04.

Question 10

A person is asked to state his age in years. He says, "Take my age three years from now, multiply it by three, then subtract three times my age three years ago, and you will get my present age." How old is the person now?

Accepted Answer

Introduction / Context: This is an algebraic age puzzle that requires translating a verbal description into a simple equation. Such problems are common in aptitude exams because they test logical interpretation of language as well as basic linear equation solving skills. Given Data / Assumptions: Let the present age of the person be A years. Age three years from now is A + 3. Age three years ago was A - 3. The statement says: three times age three years from now minus three times age three years ago equals the present age A. Concept / Approach: The strategy is to convert the verbal condition into an algebraic equation. Once we write the equation in terms of A, we can simplify it and solve for A using basic algebra. The structure of the statement sets up a linear equation with only one unknown, which is straightforward to solve. Step-by-Step Solution: Let the present age be A years.Age three years from now = A + 3.Age three years ago = A - 3.Given condition: 3 × (A + 3) - 3 × (A - 3) = A.Expand both products: 3(A + 3) = 3A + 9 and 3(A - 3) = 3A - 9.Substitute into the equation: (3A + 9) - (3A - 9) = A.Simplify the left side: 3A + 9 - 3A + 9 = 18.So we get 18 = A.Therefore, the present age of the person is 18 years. Verification / Alternative check: Check the statement using A = 18. Age three years from now: 18 + 3 = 21. Age three years ago: 18 - 3 = 15. Now compute 3 × 21 - 3 × 15 = 63 - 45 = 18, which matches the present age. The condition is satisfied exactly, confirming that 18 years is the correct solution. No other age will satisfy this specific equation. Why Other Options Are Wrong: If A were 20, then 3(A + 3) - 3(A - 3) would be 3 × 23 - 3 × 17 = 69 - 51 = 18, not 20. For 24, the expression gives 3 × 27 - 3 × 21 = 81 - 63 = 18 again, not 24. Similar checks for 30 or 36 show that the left side remains 18 because the A terms cancel out, so only A = 18 fulfills the required equality. Common Pitfalls: Some learners misinterpret the phrases three years hence and three years ago and may write A + 3 and 3 - A or mix up the direction of time. Others forget to distribute the multiplication correctly over the parentheses, leading to incorrect expansion. Careful reading of the statement and stepwise algebraic expansion help avoid such mistakes. Final Answer: The person is currently 18 years old.

Question 11

Which of the following sequences of fractions is correctly arranged in descending order of their values (from largest to smallest)?

Accepted Answer

Introduction / Context: This question tests your ability to compare fractions and arrange them in descending order. Comparing fractions with different denominators is a core skill in arithmetic and is important for number sense, ratio problems, and probability questions. Given Data / Assumptions: The fractions to be compared are 5/9, 7/11, 8/15, and 11/17. We must find the sequence that lists these fractions from largest to smallest. All fractions are positive and less than 1. The options provide different possible orderings of these four fractions. Concept / Approach: There are two standard approaches: convert each fraction into a decimal approximation or compare pairs using cross multiplication. Because the denominators are different and not simple multiples of each other, decimal approximations or careful cross multiplication are convenient. Our goal is to determine which fraction is largest, which is smallest, and the correct order in between. Step-by-Step Solution: Approximate each fraction as a decimal.5/9 ≈ 0.5556.7/11 ≈ 0.6364.8/15 ≈ 0.5333.11/17 ≈ 0.6471.Now rank these approximate decimal values from largest to smallest: 0.6471 (11/17), 0.6364 (7/11), 0.5556 (5/9), 0.5333 (8/15).Thus the correct descending order is 11/17, 7/11, 5/9, 8/15. Verification / Alternative check: We can also verify using cross multiplication without decimals. To compare 11/17 and 7/11, compute 11 × 11 = 121 and 7 × 17 = 119. Since 121 > 119, 11/17 > 7/11. Compare 7/11 and 5/9: 7 × 9 = 63 and 5 × 11 = 55, so 7/11 > 5/9. Compare 5/9 and 8/15: 5 × 15 = 75 and 8 × 9 = 72, so 5/9 > 8/15. These comparisons confirm the exact same ordering: 11/17 > 7/11 > 5/9 > 8/15. Why Other Options Are Wrong: Any option that places a smaller fraction before a larger one is incorrect. For example, options that start with 5/9 or 7/11 ignore that 11/17 is actually greater than both. Options that place 8/15 before 5/9 contradict our comparison showing 5/9 is larger. Only the option listing 11/17, 7/11, 5/9, 8/15 respects all the pairwise comparisons and therefore represents the true descending order. Common Pitfalls: Some learners try to compare fractions by looking only at numerators or denominators, which is misleading when denominators differ. Others may attempt to convert only some fractions to decimals and misjudge the relative sizes due to rounding. Using systematic cross multiplication or consistent decimal approximations for all fractions avoids these issues and leads to the correct ordering. Final Answer: The correct descending order of the given fractions is 11/17, 7/11, 5/9, 8/15.

Question 12

When 52416 is divided by 312, the quotient is 168. Using this information, what will be the quotient when 52.416 is divided by 0.0168?

Accepted Answer

Introduction / Context: This question tests your understanding of how scaling both the dividend and divisor by powers of 10 affects the quotient. It shows how known division facts can be used to quickly evaluate related divisions with decimal numbers, a common trick in aptitude tests to save time. Given Data / Assumptions: We know that 52416 ÷ 312 = 168. We must find the value of 52.416 ÷ 0.0168. Both division problems involve the same digits but with different decimal placements. We assume all computations are exact, with no rounding. Concept / Approach: The key idea is that multiplying or dividing both the numerator and denominator of a fraction by the same nonzero number does not change its value. We represent 52.416 ÷ 0.0168 as a fraction and then scale numerator and denominator by a power of 10 so that we can relate it back to 52416 ÷ 312. By counting decimal places, we can see how to transform one expression into the other. Step-by-Step Solution: Write the required quotient as a fraction: 52.416 ÷ 0.0168 = 52.416 / 0.0168.Count decimal places: 52.416 has three decimal places, 0.0168 has four.Multiply numerator and denominator by 10,000 to clear decimals: (52.416 × 10,000) / (0.0168 × 10,000).Compute: 52.416 × 10,000 = 524,160 and 0.0168 × 10,000 = 168.So 52.416 / 0.0168 = 524,160 / 168.Notice that 524,160 / 168 = (52416 × 10) / 168.We know 52416 ÷ 312 = 168, which implies 52416 = 168 × 312.Now, 524,160 / 168 = (168 × 312 × 10) / 168 = 312 × 10 = 3120.Thus, 52.416 ÷ 0.0168 = 3120. Verification / Alternative check: You can also verify on a calculator or by simplified division. Since 0.0168 is very small, dividing 52.416 by it should give a large number. Direct computation produces 3120, confirming our result. The relationship with 52416 ÷ 312 helps to see the connection of digits and provides a quick mental confirmation of the magnitude of the answer. Why Other Options Are Wrong: Option 3.12 or 31.2 are far too small; dividing by a small number like 0.0168 should increase the value substantially. Option 312 is closer in scale but still smaller by a factor of 10 than the correct answer. Option 168 is simply the known quotient from a different division problem and does not correspond to the new combination of decimal shifts. Only 3120 is consistent with both the scaling logic and the known division fact. Common Pitfalls: A common error is to move the decimal in the numerator and denominator by different numbers of places, which changes the value of the fraction. Another mistake is to assume that the new quotient must be the same as the old one because the digits look similar. Properly tracking the total shifts in place value and maintaining equal scaling for numerator and denominator prevents these errors. Final Answer: The quotient when 52.416 is divided by 0.0168 is 3120.

Question 13

Evaluate the expression (0.2 × 0.2 + 0.01) ÷ (0.1 × 0.1 + 0.02) and express the result as a simplified fraction.

Accepted Answer

Introduction / Context: This question involves evaluation of an expression containing decimals in both the numerator and denominator. It tests your ability to perform decimal multiplication and addition correctly and then convert the resulting ratio into a simplified fraction. Such manipulation is common in decimal fraction and algebra sections of aptitude tests. Given Data / Assumptions: The expression is (0.2 × 0.2 + 0.01) ÷ (0.1 × 0.1 + 0.02). All operations inside the numerator and denominator are standard multiplication and addition. We must evaluate the expression exactly and then express the result as a fraction in simplest form. No rounding is needed because all decimals are finite with a small number of digits. Concept / Approach: The plan is to calculate the numerator and denominator separately as decimals, then form their quotient. Once we have the decimal result, we convert it into a fraction and simplify. Because the numbers are small and neat, each step involves straightforward calculations and leads to a simple rational number. Step-by-Step Solution: First evaluate the numerator: 0.2 × 0.2 + 0.01.Compute 0.2 × 0.2 = 0.04.Add 0.01: numerator = 0.04 + 0.01 = 0.05.Next evaluate the denominator: 0.1 × 0.1 + 0.02.Compute 0.1 × 0.1 = 0.01.Add 0.02: denominator = 0.01 + 0.02 = 0.03.Now form the ratio: 0.05 ÷ 0.03.Write 0.05 as 5/100 and 0.03 as 3/100.Then 0.05 ÷ 0.03 = (5/100) ÷ (3/100) = (5/100) × (100/3) = 5/3.Therefore, the value of the expression is 5/3. Verification / Alternative check: We can also treat 0.05 ÷ 0.03 directly as a decimal division. Calculate 0.05 ÷ 0.03 by multiplying numerator and denominator by 100: 5 ÷ 3. This gives 1.6666..., whose exact fractional representation is 5/3. Both the fraction manipulation method and the direct decimal division lead to the same result, confirming that 5/3 is correct. Why Other Options Are Wrong: Option 9/5 equals 1.8, which would require the numerator and denominator to be in a different ratio than 5 to 3. Option 41/4 is 10.25, far too large for a quotient of two small decimals less than 1. Option 41/12 is about 3.4167, also too large. Option 3/5 equals 0.6, which would correspond to a smaller ratio than 5/3. None of these match the computed decimal 1.6666..., so they are incorrect. Common Pitfalls: Errors often arise from miscalculating the initial decimal products, for example confusing 0.2 × 0.2 with 0.2 or 0.04 with 0.004. Another mistake is incorrectly handling the decimal division and not clearing the decimals in numerator and denominator properly. Converting each decimal to a fraction and simplifying step by step is a reliable way to avoid such mistakes. Final Answer: The simplified value of the expression (0.2 × 0.2 + 0.01) ÷ (0.1 × 0.1 + 0.02) is 5/3.

Question 14

What is the decimal representation of the fraction 1/8, written as an exact terminating decimal value?

Accepted Answer

Introduction / Context: Fractions and decimals are two different notations for the same rational numbers. This question checks whether you can convert a simple fraction with denominator eight into its exact decimal form, which is a common skill in aptitude tests, banking, and everyday calculations. Given Data / Assumptions: Given fraction: 1/8. We need to write 1/8 as a decimal. We work in base 10 decimals and do exact division. No rounding is required if the decimal terminates. Concept / Approach: To convert a fraction to a decimal, we divide the numerator by the denominator using long division or known fraction to decimal pairs. Here the numerator is 1 and the denominator is 8, so we compute 1 / 8. Since 8 does not divide 1 directly, we introduce decimal places and treat 1.000 as the dividend. This is a standard method for fraction to decimal conversion. Step-by-Step Solution: Step 1: Write the fraction as division: 1/8 means 1 divided by 8. Step 2: Since 8 does not go into 1, write 1.000 and perform division with decimals. Step 3: 8 goes into 10 one time, so the first decimal digit is 1 and the remainder is 10 - 8 = 2. Step 4: Bring down the next 0 to get 20. Now 8 goes into 20 two times, remainder 20 - 16 = 4. The next digit is 2. Step 5: Bring down another 0 to get 40. Now 8 goes into 40 five times, remainder 40 - 40 = 0, so the next digit is 5. Step 6: The division ends with zero remainder, so 1/8 = 0.125 exactly. Verification / Alternative check: We can quickly verify the result by multiplying back. If 1/8 = 0.125 is correct, then 0.125 * 8 should give 1. Compute 0.125 * 8 = 1.000, which matches the original numerator. Therefore the decimal 0.125 is correct and exact. We can also recall a standard table of common fractions, where 1/2 = 0.5, 1/4 = 0.25, and 1/8 = 0.125. Why Other Options Are Wrong: 0.225 is greater than 0.2 and would correspond to 9/40, not 1/8. 0.325 is larger still and does not equal 1/8 when multiplied by 8. 0.425 is even larger and clearly does not give 1 when multiplied by 8. Common Pitfalls: Students sometimes guess values like 0.25 or 0.2 because they remember related fractions such as 1/4 or 1/5. Another common mistake is to stop the long division too early or to misplace the decimal point, which leads to incorrect results. Remember that 1/8 is smaller than 1/4, so its decimal must be smaller than 0.25. Careful step-by-step division avoids these errors. Final Answer: The decimal representation of the fraction 1/8 is 0.125.

Question 15

What is the decimal representation of the fraction 5/8, expressed as a terminating decimal number?

Accepted Answer

Introduction / Context: Being able to convert a fraction into a decimal is an essential arithmetic skill. This question asks you to express the fraction 5/8 as an exact decimal, which is useful in percentage calculations, measurements, and financial problems. Given Data / Assumptions: Fraction given: 5/8. We require its decimal form without rounding errors. We use standard base 10 decimal representation. Concept / Approach: To convert 5/8 into a decimal, we divide 5 by 8. Another approach is to use a known simpler fraction. If we already know that 1/8 = 0.125, then 5/8 is simply five times that decimal. Both ways rely on the idea that a fraction a/b can always be written as a decimal by performing a ÷ b, although some fractions may produce repeating decimals. Here the denominator is a power of 2, so the decimal will terminate. Step-by-Step Solution: Step 1: Recall or compute 1/8. From the previous problem, 1/8 = 0.125. Step 2: Since 5/8 = 5 * (1/8), multiply the decimal 0.125 by 5. Step 3: Compute 0.125 * 5. First 5 * 125 = 625. Then place the decimal point three places from the right to get 0.625. Step 4: Therefore, 5/8 = 0.625 as a decimal. Step 5: Alternatively, do the long division 5 ÷ 8 to reach the same result. Verification / Alternative check: To verify, multiply the decimal 0.625 back by 8. Compute 0.625 * 8 = 5.000, which matches the original numerator. This confirms that 0.625 is correct. Since 5/8 is more than one half (0.5) and less than three fourths (0.75), its decimal should lie between 0.5 and 0.75, and 0.625 fits this range perfectly. Why Other Options Are Wrong: 0.541 is less than 0.625 and does not give 5 when multiplied by 8. 0.258 is far too small and would correspond to roughly 1/4, not 5/8. 0.147 is even smaller and clearly cannot represent 5/8, which is greater than one half. Common Pitfalls: Learners sometimes confuse 3/8 and 5/8, or they guess decimals such as 0.58 based on the numerator. Another mistake is failing to use a known value like 1/8 to speed up the computation. Always confirm by multiplying the decimal by the denominator to see if the original numerator is recovered. Final Answer: The decimal representation of 5/8 is 0.625.

Question 16

When the fraction 125 over 1000 is reduced to simplest form, what is the resulting fraction?

Accepted Answer

Introduction / Context: This question tests your understanding of simplifying fractions to their lowest terms. Simplifying is important because it gives a standard, easiest to read form for a rational number, and many later calculations become more convenient when fractions are reduced first. Given Data / Assumptions: Original fraction: 125/1000. We must find its simplest or lowest terms form. We assume standard fraction simplification using common factors. Concept / Approach: To simplify a fraction, we divide both numerator and denominator by their greatest common divisor. Here, 125 and 1000 are both multiples of 125. Dividing top and bottom by the same non zero number does not change the value of the fraction, it only changes its appearance to a simpler form. This is the central idea of fraction simplification. Step-by-Step Solution: Step 1: Start with the fraction 125/1000. Step 2: Find a common factor of 125 and 1000. Clearly 125 divides both numbers. Step 3: Divide the numerator by 125: 125 / 125 = 1. Step 4: Divide the denominator by 125: 1000 / 125 = 8. Step 5: The simplified fraction is 1/8. Step 6: Check whether 1 and 8 have any further common factor greater than 1. They do not, so 1/8 is in simplest form. Verification / Alternative check: We can verify with decimal equivalents. 125/1000 equals 0.125, because 125 over 1000 represents 125 thousandths. The fraction 1/8 also equals 0.125 as shown in an earlier question. Since 125/1000 and 1/8 have the same decimal value, they represent the same number, and 1/8 is clearly in lowest terms. Why Other Options Are Wrong: 2/9 is approximately 0.222 and does not equal 0.125. 3/8 is 0.375, which is three times larger than 1/8. 4/7 is a non terminating decimal around 0.571 and definitely not equal to 125/1000. Common Pitfalls: A common error is to divide by a small factor like 5 or 25 and stop too early, leaving the fraction not fully simplified. Another mistake is to miscalculate the division of the denominator, leading to incorrect fractions. Always check whether the resulting numerator and denominator have any further common factor other than 1. Final Answer: The simplest form of 125 over 1000 is 1/8.

Question 17

Evaluate the decimal expression (0.2 × 0.2 + 0.01) divided by (0.1 × 0.1 + 0.02), and express the result as a simplified fraction.

Accepted Answer

Introduction / Context: This problem checks your ability to work with decimal operations and algebraic structure. The expression contains products and sums of decimals in both the numerator and denominator, and the final answer needs to be expressed as a simplified fraction. Such questions are common in aptitude tests to see if you can handle precise decimal arithmetic and recognize patterns. Given Data / Assumptions: Numerator: 0.2 × 0.2 + 0.01. Denominator: 0.1 × 0.1 + 0.02. We must compute (numerator) / (denominator). All values are exact decimals, and we assume standard arithmetic rules. Concept / Approach: First we simplify the numerator and denominator separately by performing multiplication and addition. Once both parts are simplified to single decimal numbers, we divide the numerator by the denominator. When possible, we then convert the resulting decimal into a fraction and simplify that fraction. Recognizing that 0.2 = 2/10 and 0.1 = 1/10 can also help to see the fraction relationships more clearly. Step-by-Step Solution: Step 1: Compute the numerator: 0.2 × 0.2 = 0.04. Step 2: Add 0.01 to the numerator: 0.04 + 0.01 = 0.05. Step 3: Compute the denominator: 0.1 × 0.1 = 0.01. Step 4: Add 0.02 to the denominator: 0.01 + 0.02 = 0.03. Step 5: The expression becomes 0.05 / 0.03. Step 6: Write 0.05 as 5/100 and 0.03 as 3/100. Step 7: Now 0.05 / 0.03 = (5/100) / (3/100) = (5/100) × (100/3) = 5/3. Verification / Alternative check: We can also treat 0.05 / 0.03 as a direct decimal division. 0.05 divided by 0.03 equals 1.6666 recurring, which is the decimal form of 5/3. Checking the fraction by converting 5/3 to decimal gives 1.6666 recurring, confirming alignment. Both methods support the same result, so our answer is consistent and reliable. Why Other Options Are Wrong: 9/3 simplifies to 3, which is much larger than the computed value near 1.67. 11/3 is approximately 3.67, again far from the correct ratio. 13/3 is about 4.33, clearly not matching 0.05 / 0.03. Common Pitfalls: A common mistake is to add or multiply decimals incorrectly, such as misplacing decimal points. Some learners also forget that dividing by a decimal can be simplified by converting to fractions or by scaling both numerator and denominator by 100. Always calculate products first, then sums, and only finally perform the division. Final Answer: The value of the expression is 5/3.

Question 18

Evaluate the expression (3.39)^2 − (2.69)^2 and choose the nearest integer value of the result.

Accepted Answer

Introduction / Context: This question involves squaring decimal numbers and then subtracting those squares. It checks your comfort with decimal arithmetic and also allows the use of an algebraic identity for a faster solution. Such problems are common in aptitude exams where accuracy with decimals is crucial. Given Data / Assumptions: We need to evaluate (3.39)^2 − (2.69)^2. We are asked to select the nearest integer value from the given options. All calculations use standard decimal arithmetic. Concept / Approach: Instead of squaring both numbers directly, we can use the algebraic identity for difference of squares. The identity is a^2 − b^2 = (a − b) * (a + b). Applying this identity with a = 3.39 and b = 2.69 avoids long multiplication of decimals and greatly simplifies the work. Step-by-Step Solution: Step 1: Identify a and b. Here a = 3.39 and b = 2.69. Step 2: Use the identity a^2 − b^2 = (a − b) * (a + b). Step 3: Compute a − b = 3.39 − 2.69 = 0.70. Step 4: Compute a + b = 3.39 + 2.69 = 6.08. Step 5: Now compute the product: (a − b) * (a + b) = 0.70 * 6.08. Step 6: 0.70 * 6.08 = 4.256. Step 7: The exact result is 4.256, which is closest to the integer 4 among the given options. Verification / Alternative check: As a verification, we can roughly estimate. 3.39 is just under 3.4 and 2.69 is just under 2.7. Using approximate values, (3.4)^2 = 11.56 and (2.7)^2 = 7.29, so the difference is about 4.27, which is also closest to 4. This agrees with the more accurate calculation of 4.256. Therefore option 4 is confirmed as the correct nearest integer. Why Other Options Are Wrong: 3 is farther from 4.256 than 4 is, so it is not the nearest integer. 6 is significantly larger than 4.256 and does not match the computed value. 1 is far too small when compared with the exact result. Common Pitfalls: Many learners try to square the decimals directly and may make mistakes with decimal places. Others forget the difference of squares identity and therefore perform longer calculations than necessary. Some might also misinterpret the question and subtract the numbers first and then square, which would give (3.39 − 2.69)^2 instead of a^2 − b^2. Always read the expression carefully and apply the correct algebraic identity. Final Answer: The value of (3.39)^2 − (2.69)^2 is approximately 4.256, so the nearest integer is 4.

Question 19

What is the vulgar fraction (ordinary fraction) that is exactly equal to the decimal 0.75?

Accepted Answer

Introduction / Context: This question checks your ability to convert a simple terminating decimal into an equivalent fraction, sometimes called a vulgar fraction. Converting between decimals and fractions is important in topics like percentages, ratios, and measurement conversions. Given Data / Assumptions: Given decimal: 0.75. We need to express 0.75 as a fraction in simplest form. We assume base 10 decimals and exact arithmetic without rounding. Concept / Approach: A terminating decimal can be written as a fraction whose denominator is a power of 10. After that, we simplify the fraction by dividing numerator and denominator by their greatest common divisor. Here, 0.75 has two decimal places, so it can initially be written over 100 and then reduced. Step-by-Step Solution: Step 1: Write 0.75 as a fraction with denominator 100. Since there are two decimal places, 0.75 = 75/100. Step 2: Find the greatest common divisor of 75 and 100. Both are divisible by 25. Step 3: Divide numerator and denominator by 25. We get 75 / 25 = 3 and 100 / 25 = 4. Step 4: Therefore 0.75 = 3/4 in simplest form. Step 5: Check whether 3 and 4 have any further common factor greater than 1. They do not, so 3/4 is fully simplified. Verification / Alternative check: We can verify by converting 3/4 back to a decimal. Compute 3/4 as 3 ÷ 4. Since 4 goes into 3.00 exactly 0.75 times, the decimal equals 0.75. This confirms that the original decimal and the fraction 3/4 represent the same value. Why Other Options Are Wrong: 3/5 equals 0.6, which is smaller than 0.75. 3/2 equals 1.5, which is greater than 0.75 and is an improper fraction here. 3/7 is approximately 0.4286, also not equal to 0.75. Common Pitfalls: Some learners confuse 0.75 with 0.7 or 0.8 and may incorrectly choose 3/5 or another fraction. Others may stop at 75/100 and forget to simplify. Remember that fractions should usually be given in lowest terms unless the question states otherwise. Final Answer: The vulgar fraction equal to 0.75 is 3/4.

Question 20

Among the fractions 2/3, 3/4, 4/5, and 5/6, what is the difference between the largest and the smallest fraction?

Accepted Answer

Introduction / Context: This question tests your ability to compare fractions and compute the difference between them. Such comparison skills are widely used in ratio problems, percentage growth, and many quantitative aptitude topics. Given Data / Assumptions: Fractions given: 2/3, 3/4, 4/5, and 5/6. We must find the largest fraction and the smallest fraction. Then we compute the difference: largest − smallest. Concept / Approach: To compare fractions, we can convert them to a common denominator, convert to decimals, or reason by how close they are to 1. All four fractions are less than 1 but greater than 1/2. We can use the idea that for fractions with the same numerator, a smaller denominator gives a larger fraction, and for fractions close to 1, the one with the smallest difference from 1 will be the largest. Step-by-Step Solution: Step 1: Convert each fraction to decimal or compare directly. Step 2: 2/3 is approximately 0.6667. Step 3: 3/4 equals 0.75. Step 4: 4/5 equals 0.8. Step 5: 5/6 is approximately 0.8333. Step 6: From these values, the smallest fraction is 2/3 and the largest fraction is 5/6. Step 7: Compute the difference: (5/6) − (2/3). First rewrite 2/3 with denominator 6: 2/3 = 4/6. Step 8: Now 5/6 − 4/6 = 1/6. Verification / Alternative check: We can check the difference numerically using decimals. 5/6 is about 0.8333 and 2/3 is about 0.6667. The difference is roughly 0.1666, which is the decimal value of 1/6. This confirms our fraction calculation. Also, checking with a common denominator of 60 would give 2/3 = 40/60 and 5/6 = 50/60, and again the difference is 10/60 = 1/6. Why Other Options Are Wrong: 1/12 is approximately 0.0833, too small compared to the computed difference. 1/20 equals 0.05, also much smaller than the actual difference. 1/30 is about 0.0333, which is far below the true gap between 2/3 and 5/6. Common Pitfalls: One common mistake is to assume that 2/3 is larger than 3/4 because the numerator is larger, which is incorrect. Another error is to subtract numerators and denominators directly, which is not valid for fraction subtraction. Always convert to a common denominator or to decimals when comparing or subtracting fractions. Final Answer: The difference between the largest and smallest fractions is 1/6.

Decimal Fraction Questions