Question 1

Divide small decimals accurately Evaluate: 0.000033 ÷ 0.11 = ?

Accepted Answer

Introduction / Context: Dividing very small decimals is easiest when you convert to fraction or scientific notation. This prevents place-value mistakes and makes mental checks straightforward. Given Data / Assumptions: Compute 0.000033 / 0.11. Exact decimal values are intended; no rounding required. Concept / Approach: Use scientific notation: 0.000033 = 3.3 * 10^-5 and 0.11 = 1.1 * 10^-1. Then divide coefficients and subtract exponents. Alternatively, scale both numerator and denominator by 100 to remove decimals judiciously. Step-by-Step Solution:Write 0.000033 = 3.3 × 10^-5 and 0.11 = 1.1 × 10^-1.Divide: (3.3 / 1.1) × 10^(-5 − (−1)) = 3 × 10^-4.Convert to decimal: 3 × 10^-4 = 0.0003. Verification / Alternative check: As fractions, 0.000033 / 0.11 = (33/1,000,000) / (11/100) = (33/1,000,000) * (100/11) = 3/10000 = 0.0003. Matches the previous method. Why Other Options Are Wrong:.003, .03, .3: Each is 10, 100, or 1000 times too large, reflecting misplaced decimal points. Common Pitfalls: Moving the decimal in only the numerator or denominator; forgetting that division by 0.11 should increase the magnitude modestly but not drastically. Scientific notation helps keep track of powers of 10. Final Answer: 0.0003

Question 2

Large result from dividing by a small decimal Evaluate: 25 ÷ 0.0005 = ?

Accepted Answer

Introduction / Context: Dividing by a very small decimal yields a large result. This problem reinforces scale intuition and careful handling of zeros and decimal places to avoid underestimating the quotient. Given Data / Assumptions: Compute 25 / 0.0005. No rounding; the value is exact. Concept / Approach: Convert 0.0005 to a fraction for clarity. Since 0.0005 = 5/10000 = 1/2000, dividing by 0.0005 is equivalent to multiplying by 2000, which makes mental computation straightforward. Step-by-Step Solution:0.0005 = 1/2000.Therefore, 25 / 0.0005 = 25 / (1/2000) = 25 * 2000 = 50000. Verification / Alternative check: Multiply denominator and numerator by 10000 to remove decimals: (25 * 10000) / (0.0005 * 10000) = 250000 / 5 = 50000. Same result. Why Other Options Are Wrong:50, 500, 5000: Each is off by a factor of 1000, 100, or 10 respectively; they underestimate the effect of dividing by a very small number. Common Pitfalls: Treating 0.0005 as 1/200 instead of 1/2000; dropping zeros when scaling; or assuming division by a small decimal yields only a slightly larger number. Always convert to a simple fraction to keep scale consistent. Final Answer: None of these (the correct value is 50000).

Question 3

Brackets and “of” in decimal expressions Evaluate: [12 / (0.09 of 0.3)] × 2 = ?

Accepted Answer

Introduction / Context: This exercise combines bracket precedence with the word “of” meaning multiplication. Correctly evaluating inner products before division and multiplication is crucial for accurate results in compound decimal problems. Given Data / Assumptions: Expression: [12 / (0.09 of 0.3)] × 2. “of” means multiplication. Standard order: parentheses (or brackets), then division/multiplication left to right. Concept / Approach: First compute the inner product 0.09 * 0.3. Then divide 12 by that result. Finally multiply by 2. Avoid mixing integer intuition with decimal precision; keep exact values until the end. Step-by-Step Solution:Compute inner: 0.09 * 0.3 = 0.027.Divide: 12 / 0.027 = 444.444... (exactly 400/0.9 = 444.444...).Multiply by 2: 444.444... * 2 = 888.888... . Verification / Alternative check: Write as fractions: 0.09 = 9/100 and 0.3 = 3/10, so denominator = 27/1000. Then 12 / (27/1000) = 12 * (1000/27) = 12000/27 = 444.444..., and doubling gives 888.888.... Why Other Options Are Wrong:0.8, 0.08, 8: Each is orders of magnitude too small; they ignore the very small denominator 0.027 which inflates the quotient. Common Pitfalls: Treating “of” as addition or neglecting bracket precedence; rounding intermediate steps too early. Keep the exact fraction form until the final step to avoid compounding errors. Final Answer: None of these (the exact value is 888.888...).

Question 4

Scale comparison via decimal division Evaluate: 50.8 ÷ 2540 = ?

Accepted Answer

Introduction / Context: This problem checks decimal division and recognition that multiplying a divisor by 0.02 can reproduce the dividend exactly. Spotting such proportional relationships speeds up calculations on tests. Given Data / Assumptions: Compute 50.8 / 2540. No rounding is needed. Concept / Approach: Look for a simple factor k such that 2540 * k = 50.8. Since 2540 * 0.02 = 50.8, the quotient is 0.02. Alternatively, scale both numbers to remove decimals and reduce the fraction. Step-by-Step Solution:Test k = 0.02 ⇒ 2540 * 0.02 = 50.8.Hence 50.8 / 2540 = 0.02 directly. Verification / Alternative check: As a fraction: 50.8/2540 = 508/25400 = divide by 254 ⇒ 2/100 = 0.02. The simplification confirms the same result. Why Other Options Are Wrong:2, 0.2, 0.002: These correspond to multipliers that yield 5080, 508, or 5.08 when applied to 2540, not 50.8. Common Pitfalls: Dropping or adding a zero when scaling; mistaking 0.2 for 0.02. Verify by multiplying back: divisor * quotient should equal dividend exactly. Final Answer: 0.02

Question 5

Comparing two small decimals by division Evaluate: 0.0169 ÷ 0.013 = ?

Accepted Answer

Introduction / Context: Dividing close decimal values often yields a simple ratio if the numerator is a multiple of the denominator. Recognizing 0.0169 as 1.3 times 0.013 speeds up the solution. Given Data / Assumptions: Compute 0.0169 / 0.013. All values are exact decimals. Concept / Approach: Check whether 0.013 times a simple decimal like 1.3 equals 0.0169. Since 13 * 13 = 169, this becomes a straightforward mental check using scaled integers. Step-by-Step Solution:Compute 0.013 * 1.3 = (13/1000) * (13/10) = 169/10000 = 0.0169.Therefore 0.0169 / 0.013 = 1.3. Verification / Alternative check: Express as fractions: (169/10000) / (13/1000) = (169/10000) * (1000/13) = 169/130 = 1.3. Exact match. Why Other Options Are Wrong:.13 and .013: Both are much less than 1, but the numerator is larger than the denominator, so the result must exceed 1.13: Ten times too large; would imply 0.013 * 13 = 0.169, not 0.0169. Common Pitfalls: Misplacing decimal points when multiplying 13 by 13; forgetting that quotient > 1 since numerator > denominator. Validate by multiplying back. Final Answer: 1.3

Question 6

Multiply a decimal by a decimal Evaluate: 15.60 × 0.30 = ?

Accepted Answer

Introduction / Context: Multiplying decimals is a fundamental arithmetic skill. The answer can be found either by removing decimal points, multiplying integers, and then restoring decimal places, or by using fraction equivalents. Given Data / Assumptions: Compute 15.60 * 0.30. Exact arithmetic; no rounding is needed. Concept / Approach: Count total decimal places: 15.60 has two, 0.30 has two, so the product will have four decimal places when multiplying 1560 by 30, then adjust to the proper scale. Simplify trailing zeros at the end if appropriate. Step-by-Step Solution:Remove decimals: 15.60 → 1560 (2 decimal places), 0.30 → 30 (2 decimal places).Multiply integers: 1560 * 30 = 46800.Restore 4 decimal places: 46800 → 4.6800.Trim trailing zeros to standard form: 4.68. Verification / Alternative check: Fraction method: 15.60 = 1560/100 and 0.30 = 30/100, so product = (1560 * 30) / 10000 = 46800/10000 = 4.68. Same result. Why Other Options Are Wrong:0.458, 0.468, 0.0468: Each ignores the correct total of four decimal places before simplification or miscalculates the integer product. Common Pitfalls: Mixing up decimal places or dropping zeros incorrectly. Always multiply as integers first, then place the decimal using the total count of places from both factors. Final Answer: 4.68

Question 7

Decimal multiplication with powers of 10: Compute the exact value of 3 × 0.3 × 0.03 × 0.003 × 30. Show careful handling of decimal places and powers of 10 to avoid rounding errors.

Accepted Answer

Introduction / Context:This problem checks fluency with decimal multiplication and powers of 10. Multiplying several decimals often leads to misplaced decimal points. A structured approach using scientific notation prevents such mistakes and ensures an exact final answer. Given Data / Assumptions: Expression: 3 × 0.3 × 0.03 × 0.003 × 30 All quantities are exact decimals (no rounding required). Standard arithmetic precedence applies (only multiplication here). Concept / Approach:Convert decimals to powers of 10 whenever helpful. Group convenient factors to simplify mental arithmetic. Track exponents carefully, because each decimal place corresponds to a negative power of 10 (for example, 0.3 = 3 × 10^-1). Step-by-Step Solution: Group integers: 3 × 30 = 90Convert decimals: 0.3 × 0.03 × 0.003 = (3×10^-1) × (3×10^-2) × (3×10^-3)Multiply digits: 3 × 3 × 3 = 27Add exponents: 10^(-1-2-3) = 10^-6So 0.3 × 0.03 × 0.003 = 27 × 10^-6 = 0.000027Now multiply by 90: 90 × 0.000027 = 0.00243 Verification / Alternative check: Count decimal places directly: there are 1 + 2 + 3 = 6 decimal places in the three small factors; 3 × 30 = 90 scales the product by 90, giving the same 0.00243. Why Other Options Are Wrong: .0000243 and .000243: Under-count decimal shift. .0243: Over-shifts the decimal by two places. Common Pitfalls: Losing track of exponents when multiplying several decimals. Rounding mid-way instead of keeping exact values until the end. Final Answer: 0.00243

Question 8

Find the missing number in a decimal equation: Solve for ? in 16.7 + 12.38 − ? = 10.09, showing clean decimal subtraction and addition.

Accepted Answer

Introduction / Context:This is a straightforward decimal arithmetic problem. It tests accuracy in adding and subtracting decimal numbers and rearranging an equation to isolate the unknown. Such items are common in aptitude sections to assess precision and speed. Given Data / Assumptions: Equation: 16.7 + 12.38 − ? = 10.09 All values are exact decimals. The goal is to solve for ?. Concept / Approach:Rearrange the equation to isolate the unknown on one side. Use the identity a + b − x = c ⇒ x = a + b − c. Be careful with decimal alignment when adding and subtracting to avoid place-value errors. Step-by-Step Solution: Compute the sum first: 16.70 + 12.38 = 29.08Now subtract the right-hand side: x = 29.08 − 10.09x = 18.99 Verification / Alternative check: Plug back: 16.7 + 12.38 − 18.99 = 29.08 − 18.99 = 10.09 ✔ Why Other Options Are Wrong: 17.89: Off by 1.10; likely mis-subtraction. 16.98: Incorrect aligning of decimal places. 20.09: Confuses add/subtract order. Common Pitfalls: Not aligning decimals precisely when subtracting. Dropping trailing zeros (e.g., 16.70) and miscounting place values. Final Answer: 18.99

Question 9

Add mixed-precision decimals carefully: Evaluate 0.6 + 0.66 + 0.066 + 6.606 ensuring correct place-value handling and exact total.

Accepted Answer

Introduction / Context:This problem emphasizes accurate addition of decimals having different numbers of digits after the decimal point. A reliable technique is to align decimal points vertically and add place by place to avoid small but costly errors. Given Data / Assumptions: Addends: 0.6, 0.66, 0.066, 6.606 No rounding is necessary; values are exact. Concept / Approach:Keep a consistent number of decimal places by padding with zeros as needed. That allows easy column addition. Alternatively, group numbers to make round figures, but still verify by column addition. Step-by-Step Solution: 0.6 + 0.66 = 1.261.26 + 0.066 = 1.3261.326 + 6.606 = 7.932 Verification / Alternative check: Column method: write as 0.600 + 0.660 + 0.066 + 6.606 and add by columns to confirm 7.932. Why Other Options Are Wrong: 6.744 and 6.738: Miss more than one whole unit; likely dropped the 6.606 contribution properly. 7.388: Typical carry/column slip when adding 0.66 and 0.066. Common Pitfalls: Failing to align decimals precisely. Rounding mid-calculation instead of at the end (not required here). Final Answer: 7.932

Question 10

Time–decimal conversion: What decimal fraction of one hour is one second? (Choose the closest option to five decimal places.)

Accepted Answer

Introduction / Context:Converting between time units and decimal fractions is common in data entry, time sheets, and physics problems. Here, we convert 1 second into the decimal part of one hour and then match to the nearest option provided. Given Data / Assumptions: 1 hour = 60 minutes = 3600 seconds Quantity of interest: 1 second as a fraction of 1 hour Select the closest value among the options. Concept / Approach:The decimal fraction of an hour represented by 1 second is 1/3600. Converting the fraction to a decimal ensures comparability with the choices. Since options are rounded, choose the closest to the true decimal value. Step-by-Step Solution: Fraction: 1 second / 1 hour = 1 / 3600Decimal: 1 / 3600 ≈ 0.000277777…To five decimal places, ≈ 0.00028 (exact rounding). Among the given options, the closest listed is 0.00027. Verification / Alternative check: Check absolute difference: |0.00028 − 0.00027| = 0.00001, which is minimal compared with other options. Why Other Options Are Wrong: .0025 and .0256: Correspond to much larger fractions of an hour (minutes), not seconds. .000126: Too small (≈ 1/7937 hours), not equal to one second. Common Pitfalls: Confusing 1/3600 with 1/3660 or 1/360. Misplacing the decimal point when converting small fractions. Final Answer: 0.00027 (closest among options; true value ≈ 0.00028)

Question 11

Compare fractions to find the smallest value: Among 11/13, 9/11, 3/4, and 5/7, determine which fraction is the smallest.

Accepted Answer

Introduction / Context:Comparing fractions appears frequently in aptitude tests. Efficient techniques include converting to decimals mentally, using cross-multiplication for pairs, or bringing them to a common denominator. Here we identify the smallest among four familiar ratios. Given Data / Assumptions: Candidates: 11/13, 9/11, 3/4, 5/7 All are proper fractions between 0 and 1. Concept / Approach:Two quick approaches: approximate decimal values or cross-multiply pairwise against a benchmark. Noting that denominators near numerators give fractions close to 1; larger gaps yield smaller values. Compute or estimate to rank them reliably. Step-by-Step Solution: 11/13 ≈ 0.84629/11 ≈ 0.81823/4 = 0.755/7 ≈ 0.7143Smallest ≈ 0.7143 ⇒ 5/7 Verification / Alternative check: Cross-compare 5/7 vs 3/4: 5*4 = 20; 3*7 = 21 ⇒ 20 < 21 so 5/7 < 3/4. Similarly check against 9/11 and 11/13 to confirm. Why Other Options Are Wrong: 11/13, 9/11, 3/4: Each has a value greater than 5/7 when compared by cross-multiplication or decimal conversion. Common Pitfalls: Assuming larger denominators always imply smaller values without considering numerators. Rounding errors when estimating decimals too coarsely. Final Answer: 5/7

Question 12

Evaluate a decimal ratio precisely: Compute (.24 × .35) / (.14 × .15 × .02) and select the exact value.

Accepted Answer

Introduction / Context:Division of decimal products is easiest when we remove decimals by powers of 10 or compute numerically with careful place-value handling. This example keeps numbers small and clean, ideal for mental or quick written calculation. Given Data / Assumptions: Numerator: 0.24 × 0.35 Denominator: 0.14 × 0.15 × 0.02 Concept / Approach:Compute numerator and denominator separately, then divide. Alternatively, scale all terms by a sufficient power of 10 to clear decimals and simplify before division (the ratio remains unchanged). Step-by-Step Solution: Numerator: 0.24 × 0.35 = 0.084Denominator: 0.14 × 0.15 = 0.021; then 0.021 × 0.02 = 0.00042Quotient: 0.084 / 0.00042 = 200 Verification / Alternative check: Scale by 10,000: (2400 × 3500) / (1400 × 1500 × 200) simplifies by cancellations to 200. Why Other Options Are Wrong: 2, 20, 2000: Each corresponds to a misplacement of decimal points or a missed zero during scaling. Common Pitfalls: Losing track of total decimal places when multiplying three small decimals. Not simplifying common factors after clearing decimals. Final Answer: 200

Question 13

Scale a known quotient to a new one: Given 2805 ÷ 2.55 = 1100, evaluate 280.5 ÷ 25.5 by using proportional scaling rather than long division.

Accepted Answer

Introduction / Context:When a related quotient is known, you can exploit scaling to find a new quotient quickly. This avoids redundant long division and is a powerful test shortcut in decimal-fraction topics. Given Data / Assumptions: Known: 2805 ÷ 2.55 = 1100 Find: 280.5 ÷ 25.5 Concept / Approach:Observe that the new numerator is 1/10 of 2805 and the new denominator is 10 times 2.55. Since division is multiplicative, (a/10) ÷ (10b) = (a/b) ÷ 100. Thus the new quotient equals the old quotient divided by 100. Step-by-Step Solution: Original quotient Q = 2805 / 2.55 = 1100New quotient = (2805/10) / (2.55×10) = (2805 / 2.55) × (1/100)= 1100 / 100 = 11 Verification / Alternative check: Compute directly: 280.5 ÷ 25.5 = (2805 ÷ 255) = 11 (after moving decimals equally). Why Other Options Are Wrong: 1.01, 1.1, 0.11: These represent dividing by 1000, 10, or 10,000 incorrectly; the correct scaling factor is 100. Common Pitfalls: Not applying equal decimal shifts to numerator and denominator. Mishandling powers of 10 while rescaling. Final Answer: 11

Question 14

Reciprocal scaling with powers of 10: Given 1 ÷ 3.718 = 0.2689, find 1 ÷ 0.0003718 by relating the two denominators via powers of 10.

Accepted Answer

Introduction / Context:Reciprocals of decimals are often connected by scaling denominators with powers of 10. Recognizing and using this relationship converts a new reciprocal into a simple multiple of a known one, saving time and effort. Given Data / Assumptions: Known: 1 / 3.718 = 0.2689 Find: 1 / 0.0003718 Concept / Approach:Note that 0.0003718 = 3.718 × 10^-4. For any nonzero x and integer k, 1 / (x × 10^-k) = (1 / x) × 10^k. Apply this with x = 3.718 and k = 4. Step-by-Step Solution: 0.0003718 = 3.718 × 10^-41 / 0.0003718 = (1 / 3.718) × 10^4= 0.2689 × 10000 = 2689 Verification / Alternative check: Direct calculator thinking: moving the decimal in the denominator four places right multiplies the reciprocal by 10^4. Why Other Options Are Wrong: 2.689: Misses the ×10^4 factor; only ×10^1. 26890: Over-scales by ×10^5. .2689: Uses the original reciprocal without scaling. Common Pitfalls: Mistaking how many places the decimal shifts (four here). Applying the scaling to the numerator instead of to the reciprocal value. Final Answer: 2689

Question 15

Solve for the missing value in a rational equation: Find ? such that (20 + 8 × 0.5) / (20 − ?) = 12.

Accepted Answer

Introduction / Context:This item assesses linear equation solving with decimals inside a fraction. The key is to simplify the numerator first, then isolate the denominator expression and solve for the unknown while preserving equality. Given Data / Assumptions: (20 + 8 × 0.5) / (20 − x) = 12 All arithmetic is exact; no rounding required. Concept / Approach:Compute the numerator cleanly. Then, cross-multiply (since the denominator is nonzero) to solve for x. Check that the solution does not make the denominator zero and that it satisfies the original equation. Step-by-Step Solution: Compute numerator: 8 × 0.5 = 4, so 20 + 4 = 24Equation: 24 / (20 − x) = 12Invert both sides: 20 − x = 24 / 12 = 2x = 20 − 2 = 18 Verification / Alternative check: Plug back: (20 + 4) / (20 − 18) = 24 / 2 = 12 ✔ Why Other Options Are Wrong: 8 or 2: Do not satisfy 24 / (20 − x) = 12. None these: Not applicable because x = 18 works exactly. Common Pitfalls: Forgetting to simplify the numerator before solving. Arithmetic slips when subtracting from 20. Final Answer: 18

Question 16

Nested division with decimals: Evaluate 0.15 ÷ (0.5 ÷ 15) by simplifying the inner division first and then completing the outer operation.

Accepted Answer

Introduction / Context:Nested divisions can be simplified by working from the inside out. This problem also reinforces that dividing by a fraction is equivalent to multiplying by its reciprocal, a crucial concept in decimal and fraction arithmetic alike. Given Data / Assumptions: Expression: 0.15 ÷ (0.5 ÷ 15) All numbers are exact decimals. Concept / Approach:First evaluate the inner operation 0.5 ÷ 15. Then divide 0.15 by that result. Alternatively, note that 0.5 ÷ 15 = 1/30, so the overall expression becomes 0.15 ÷ (1/30) = 0.15 × 30. Step-by-Step Solution: Inner: 0.5 ÷ 15 = 0.5 / 15 = 1/2 ÷ 15 = 1 / 30 ≈ 0.033333…Outer: 0.15 ÷ (1/30) = 0.15 × 30= 4.5 Verification / Alternative check: Compute directly as fractions: 0.15 = 15/100 = 3/20; then (3/20) ÷ (1/30) = (3/20) × 30 = 90/20 = 9/2 = 4.5. Why Other Options Are Wrong: 45: Treats 0.15 as 15, ignoring decimals. 0.03 or 0.45: Arise from inverting the wrong term or dropping a factor of 10. Common Pitfalls: Performing outer division first, against order of operations. Misinterpreting 0.5 ÷ 15 as 0.5 × 15. Final Answer: 4.5

Question 17

Evaluate the missing divisor in a decimal expression: (17.28 ÷ ?) ÷ (3.6 × 0.2) = 200. Determine the exact value of ? and show the steps using basic arithmetic of decimals.

Accepted Answer

Introduction / Context:This question tests decimal arithmetic and algebraic manipulation. You are given a compound expression with division and multiplication, and you must isolate the unknown divisor “?” so that the overall equality holds. Mastery of decimal operations and the correct order of operations is essential for quick problem solving in aptitude tests. Given Data / Assumptions: (17.28 ÷ ?) ÷ (3.6 × 0.2) = 200 All quantities are real numbers; standard operator precedence applies. Goal: solve for the unknown value ?. Concept / Approach:First simplify the constant product in the denominator. Then treat the remaining expression as a simple proportion. Rearranging the equation to isolate the term 17.28 ÷ ? allows solving for ? by cross-multiplication and basic division of decimals. Step-by-Step Solution:Compute 3.6 × 0.2 = 0.72.The equation becomes (17.28 ÷ ?) ÷ 0.72 = 200.So 17.28 ÷ ? = 200 × 0.72 = 144.Hence ? = 17.28 ÷ 144.Compute 17.28 ÷ 144 = 0.12 (since 1728/144 = 12 ⇒ 17.28/144 = 0.12). Verification / Alternative check:Plug ? = 0.12 back: 17.28 ÷ 0.12 = 144; 144 ÷ 0.72 = 200. The equality holds exactly, confirming the result. Why Other Options Are Wrong: 120, 12, 1.20, 1.2: These values produce the wrong intermediate quotient 17.28 ÷ ? and therefore fail to give 200 after dividing by 0.72. Common Pitfalls:Mistaking the order of division and multiplication, or miscomputing 3.6 × 0.2. Another frequent slip is moving decimals incorrectly when dividing by 144. Final Answer:0.12

Question 18

Solve the proportional equation with decimals: 3420 ÷ 19 = ( ? ÷ 0.01 ) × 7. Find the exact value of ?. Show each simplification step clearly.

Accepted Answer

Introduction / Context:This item assesses your ability to manipulate equations involving decimals and fractions. Understanding how division by a small decimal (such as 0.01) scales a number is key to solving for the unknown quickly and accurately. Given Data / Assumptions: 3420 ÷ 19 = ( ? ÷ 0.01 ) × 7 All arithmetic is exact; avoid premature rounding. We must determine the exact rational value for ?. Concept / Approach:Note that dividing by 0.01 is the same as multiplying by 100. Therefore ( ? ÷ 0.01 ) × 7 equals 700? . Reduce the equation to a simple linear form and solve for ?. Then compare with the options provided. Step-by-Step Solution:( ? ÷ 0.01 ) × 7 = 700 ?Set 3420 ÷ 19 = 700 ?.Then ? = (3420 ÷ 19) ÷ 700 = 3420 / (19 × 700).Compute 3420/700 = 171/35.Hence ? = (171/35) ÷ 19 = 171 / (35 × 19) = 9/35. Verification / Alternative check:Evaluate RHS with ? = 9/35: ( (9/35) ÷ 0.01 ) × 7 = ( (9/35) × 100 ) × 7 = (900/35) × 7 = 900/5 = 180. LHS 3420/19 = 180. The equality is satisfied. Why Other Options Are Wrong: 35/9, 18/7, 63/5, 7/18: None equals 9/35, so they do not balance the equation. None of these: Correct because the required value is 9/35, which is not listed explicitly. Common Pitfalls:Forgetting that ÷0.01 multiplies by 100; arithmetic slips when simplifying 3420/700; or not fully reducing the fraction before comparing with options. Final Answer:None of these

Question 19

Compute the decimal product accurately: 8.32 × 0.999. Use the idea that 0.999 = 1 − 0.001 for a quick evaluation.

Accepted Answer

Introduction / Context:Multiplying by 0.999 can be done very quickly by using the identity 0.999 = 1 − 0.001. This method reduces multiplication to a simple subtraction, minimizing rounding errors and mental effort. Given Data / Assumptions: Value to compute: 8.32 × 0.999 All decimals are exact; do not round prematurely. Concept / Approach:Rewrite 0.999 as 1 − 0.001. Then 8.32 × 0.999 = 8.32 × (1 − 0.001) = 8.32 − 0.00832. Simple subtraction then yields the precise product. Step-by-Step Solution:0.999 = 1 − 0.001.8.32 × 0.999 = 8.32 − (8.32 × 0.001) = 8.32 − 0.00832.Result = 8.31168. Verification / Alternative check:Standard multiplication also gives 8.31168. You can cross-check by noting that the answer must be slightly less than 8.32 (by exactly 0.00832). Why Other Options Are Wrong: 8.31668 or 8.31618: Subtracting 0.00332 or 0.00382 instead of 0.00832. 0.831168: Misplaced decimal point by one order of magnitude. 8.32168: Incorrectly adding instead of subtracting the small adjustment. Common Pitfalls:Misplacing the decimal when computing 8.32 × 0.001; adding rather than subtracting; copying errors in digits. Final Answer:8.31168

Question 20

Evaluate the quadratic identity with decimals: 9.75^2 − 2 × 9.75 × 5.75 + 5.75^2. Recognize and apply the square of a difference.

Accepted Answer

Introduction / Context:This problem directly uses the identity (a − b)^2 = a^2 − 2ab + b^2. Recognizing the pattern among the decimal numbers avoids heavy computation and leads to a quick, exact result. Given Data / Assumptions: a = 9.75, b = 5.75 Expression: a^2 − 2ab + b^2 Concept / Approach:Identify the expression as (a − b)^2. Then compute the difference (9.75 − 5.75) and square it. This is faster and less error-prone than squaring decimals separately and combining terms. Step-by-Step Solution:9.75^2 − 2 × 9.75 × 5.75 + 5.75^2 = (9.75 − 5.75)^2.Compute the difference: 9.75 − 5.75 = 4.00.Square the result: 4.00^2 = 16. Verification / Alternative check:Explicit expansion also yields 16, but the identity-based approach is cleaner. Why Other Options Are Wrong: 13.25, 12.5, 4, 3.625: These stem from arithmetic slips or misapplying identities (e.g., confusing (a − b)^2 with a^2 − b^2). Common Pitfalls:Attempting to multiply all decimals out manually, leading to copying or rounding errors. Final Answer:16

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