Question 1

The value of $(0.\overline{2} + 0.\overline{3} + 0.\overline{32})$ is

Accepted Answer

Concept & Formula To add recurring decimals, convert each individual recurring decimal into its equivalent vulgar fraction form, find a common denominator to sum them up, and then convert the final fractional sum back into a recurring decimal. Step-by-Step Solution Given expression: 0.\overline{2} + 0.\overline{3} + 0.\overline{32} 1. Convert each term to a fraction: - 0.\overline{2} = (2/9) - 0.\overline{3} = (3/9) - 0.\overline{32} = (32/99) 2. Set a common denominator (99) to add the fractions: - (2/9) = (2 × 11/9 × 11) = (22/99) - (3/9) = (3 × 11/9 × 11) = (33/99) 3. Sum the values together: (22/99) + (33/99) + (32/99) = (22 + 33 + 32/99) = (87/99) 4. Convert back to decimal form: (87/99) = 0.\overline{87} Exam Strategy & Shortcut Alternatively, expand the decimals out to a few places and add vertically: 0.222222... 0.333333... 0.323232... Sum = 0.878787... = 0.\overline{87} This expansion method lets you spot the repeating pattern 87 in under 5 seconds without calculating fractions. Common Pitfall Do not treat these as normal decimals (0.2 + 0.3 + 0.32 = 0.82), which leads directly to the trap option (b). The recurring bar signifies infinite trailing digits that alter the addition outcome. Final Answer Therefore, the correct answer is 0.\overline{87}.

Question 2

The simplification of $3.\overline{36} - 2.\overline{05} + 1.\overline{33}$ equals

Accepted Answer

Concept & Strategy When operating on multiple mixed numbers with pure recurring decimal components of the same length, group the integer parts and the fractional parts separately. This avoids converting large numbers into improper fractions. Step-by-Step Solution Given expression: 3.\overline{36} - 2.\overline{05} + 1.\overline{33} 1. Separate the whole numbers from the recurring decimal fractions: (3 + (36/99)) - (2 + (5/99)) + (1 + (33/99)) 2. Group the integer components together and the fractional components together: (3 - 2 + 1) + ((36/99) - (5/99) + (33/99)) 3. Calculate the integer sum: 3 - 2 + 1 = 2 4. Calculate the fractional sum: (36 - 5 + 33/99) = (64/99) 5. Combine the integer and fractional parts back into recurring decimal form: 2 + 0.\overline{64} = 2.\overline{64} Exam Strategy & Shortcut Since all decimal parts have a bar over exactly two digits, you can handle them as whole numbers. Add/subtract the integers: 3 - 2 + 1 = 2. Add/subtract the recurring sequences: 36 - 5 + 33 = 64. Combine them directly to get 2.\overline{64}. This takes less than 10 seconds. Common Pitfall A common trap is misaligning the subtraction, especially treating 05 as 50. Always respect the place value of the digits under the bar. Additionally, do not forget the bar on the final answer, which would incorrectly lead you to option (b). Final Answer Therefore, the correct answer is 2.\overline{64}.

Question 3

$(0.\overline{09} 	imes 7.\overline{3})$ is equal to:

Accepted Answer

Concept & Formula Multiplication involving recurring decimals cannot be done reliably in decimal form. You must strictly convert all recurring decimals into their lowest-term fraction equivalents before multiplying. Step-by-Step Solution Given expression: (0.\overline{09} × 7.\overline{3}) 1. Convert the first decimal (0.\overline{09}) to a fraction: The repeating part is 09, which is two digits long. 0.\overline{09} = (09/99) = (9/99) = (1/11) 2. Convert the second decimal (7.\overline{3}) to a fraction: Separate the integer or use the full formula. 7.\overline{3} = 7 + (3/9) = 7 + (1/3) Convert the mixed fraction to an improper fraction: ((7 × 3) + 1/3) = (22/3) 3. Multiply the two simplified fractions: (1/11) × (22/3) 4. Cancel out common factors across the numerator and denominator: The 22 in the numerator and 11 in the denominator simplify to 2. (1 × 2/1 × 3) = (2/3) 5. Convert the final fraction back to a recurring decimal to match the options: (2/3) = 0.6666... = 0.\overline{6} *(Note: The options present this as .\overline{6})* Exam Strategy & Shortcut Familiarize yourself with common fraction-to-decimal conversions. Seeing 0.\overline{09} should instantly trigger (1/11) in your mind, and 7.\overline{3} is obviously 7 (1/3) or (22/3). The multiplication (1/11) × (22/3) cleanly cancels to (2/3), and every prepared student knows (2/3) = 0.\overline{6}. Common Pitfall Attempting to multiply 0.09 × 7.3 as standard terminating decimals completely ignores the infinite nature of the numbers and leads to chaotic, incorrect results that don't match any options. Final Answer Therefore, the correct answer is .\overline{6}.

Question 4

$(0.34\overline{67} + 0.13\overline{33})$ is equal to

Accepted Answer

Concept & Formula When adding mixed recurring decimals, direct vertical addition can be tricky due to carryovers cascading infinitely. The most foolproof method is converting them into fractions, summing them, and converting the result back to a mixed recurring decimal. Fraction = \frac{Entire Number - Non-repeating part}{9s (repeating count) 0s (non-repeating count)} Step-by-Step Solution Given expression: (0.34\overline{67} + 0.13\overline{33}) 1. Convert both decimals to fractions: 0.34\overline{67} = (3467 - 34/9900) = (3433/9900) 0.13\overline{33} = (1333 - 13/9900) = (1320/9900) 2. Add the fractions: (3433/9900) + (1320/9900) = (4753/9900) 3. Convert the resulting fraction back to a decimal: Because the denominator is 9900, the final decimal has two non-repeating digits and two repeating digits, in the format 0.ab\overline{cd}. Using the formula backwards: (abcd - ab/9900) = (4753/9900) abcd - ab = 4753 Let's test ab = 48 (since the complete number is close to 4753): 48cd - 48 = 4753 48cd = 4753 + 48 48cd = 4801 This means the complete four-digit sequence is 4801. Thus, ab = 48 and cd = 01. 4. Write the final decimal: 0.48\overline{01} Exam Strategy & Shortcut You can use vertical addition if you expand carefully and watch for the carry-over from the repeating block. \quad 0.34676767... + 0.13333333... Notice 67 + 33 = 100. This causes a carry of 1 into the next block. \quad 0.48010100... The pattern 01 repeats infinitely, yielding 0.48\overline{01}. Common Pitfall A major pitfall is adding 3467 + 1333 to get 4800 and incorrectly concluding the answer is 0.48\overline{00} or just 0.48 (Option d). This ignores the fact that the infinite sequence 6767... + 3333... continuously generates carry-overs. Final Answer Therefore, the correct answer is 0.48\overline{01}.

Question 5

$(8.3\overline{1} + 0.\overline{6} + 0.00\overline{2})$ is equal to

Accepted Answer

Concept & Logic Adding decimals with different repeating patterns often creates unexpected results. When standard addition yields a terminating decimal (like 0.98), exam setters might disguise the correct option by expressing it as a recurring decimal ending in a repeating 9 (e.g., 0.97\overline{9}), which is mathematically equivalent to 0.98. Step-by-Step Solution Given expression: (8.3\overline{1} + 0.\overline{6} + 0.00\overline{2}) 1. Convert each term to a vulgar fraction: 8.3\overline{1} = 8 + (31 - 3/90) = 8 + (28/90) 0.\overline{6} = (6/9) = (60/90) 0.00\overline{2} = (2/900) 2. Find a common denominator (900) and sum them up: 8 + (28 × 10/90 × 10) + (60 × 10/90 × 10) + (2/900) 8 + (280/900) + (600/900) + (2/900) 8 + (882/900) 3. Simplify the fraction to a decimal: (882/900) = (882 \div 9/900 \div 9) = (98/100) = 0.98 So, the exact sum is 8.98. 4. Match with the given options: None of the options read exactly 8.98. However, mathematically, a number ending in an infinite sequence of 9s is exactly equal to the next terminating decimal. Let's evaluate Option (d): 8.97\overline{9} 8.97\overline{9} = 8 + (979 - 97/900) = 8 + (882/900) = 8.98 Thus, 8.97\overline{9} is mathematically identical to 8.98. Exam Strategy & Shortcut Expand the decimals to see the trend: \quad 8.311111... \quad 0.666666... + 0.002222... \rule{2.5cm}{0.4pt} \quad 8.979999... You can clearly see the sum produces 8.97\overline{9}. You don't even need to convert it to 8.98 to spot the correct option. Common Pitfall Many students reach 8.98 via fraction calculation, scan the options, panic when they don't see 8.98, and blindly guess. Recognizing that 0.97\overline{9} = 0.98 is a crucial advanced arithmetic concept. Final Answer Therefore, the correct answer is 8.97\overline{9}.

Question 6

The sum of $\overline{2}.75$ and $\overline{3}.78$ is

Accepted Answer

Concept & Logic When a bar is placed over the integer part of a decimal (the characteristic), it signifies that *only* the integer is negative, while the decimal part (the mantissa) remains positive. This notation is commonly used in logarithm calculations. \overline{a}.bc = -a + 0.bc Step-by-Step Solution Given expression: \overline{2}.75 + \overline{3}.78 1. Break down each number into its negative integer and positive decimal components: \overline{2}.75 = -2 + 0.75 \overline{3}.78 = -3 + 0.78 2. Add the equations together. Group the integer parts and the decimal parts separately: (-2 - 3) + (0.75 + 0.78) 3. Calculate the sum of the integers: -2 - 3 = -5 4. Calculate the sum of the decimals: 0.75 + 0.78 = 1.53 5. Combine the results: -5 + 1.53 Split the 1.53 into its whole number and decimal parts: -5 + 1 + 0.53 -4 + 0.53 6. Convert back into the bar notation format (negative integer, positive decimal): \overline{4}.53 Exam Strategy & Shortcut Add the decimal parts first: 0.75 + 0.78 = 1.53. This generates a +1 carry-over. Now, sum the integers along with the carry-over: -2 - 3 + 1 = -4. The integer becomes \overline{4} and the decimal remains .53, instantly giving \overline{4}.53. Common Pitfall A very common mistake is treating the entire number as negative (e.g., treating \overline{2}.75 as -2.75). This leads to (-2.75) + (-3.78) = -6.53, which might lead students to mistakenly guess an incorrect option. The bar on top is distinct from a negative sign in front. Final Answer Therefore, the correct answer is \overline{4}.53.

Question 7

If $\frac{547.527}{0.0082} = x$, then the value of $\frac{547527}{82}$ is

Accepted Answer

Concept & Logic: Shifting decimal places in a fraction alters its total value by powers of 10. By converting decimals to integers, we can track these power shifts to find the new value. Step-by-Step Solution: • Write the initial equation and convert decimals to fractions of 10: (547.527/0.0082) = x \frac{547527 × 10^{-3}}{82 × 10^{-4}} = x • Simplify the exponents: (547527/82) × 10^{(-3 - (-4))} = x (547527/82) × 10^{1} = x • Isolate the required expression: (547527/82) = (x/10) Exam Strategy & Shortcut: Count the decimal place changes from the given equation to the target equation. The numerator shifted right by 3 places, while the denominator shifted right by 4 places. The net effect on the left-hand side is a multiplication by 10. Moving this multiplier to the right side results in division by 10. Common Pitfall: Getting the sign of the power of 10 reversed during isolation, mistakenly concluding the answer is 10x. Always write out the negative exponents explicitly to avoid this. Final Answer: Therefore, the correct answer is (x/10).

Question 8

If $2994 \div 14.5 = 172$, then $29.94 \div 1.45 = x$

Accepted Answer

Concept & Strategy: When adjusting both the dividend and the divisor by decimal shifts, the overall quotient shifts by the net difference in decimal places. Step-by-Step Solution: • The given equation can be written as: (2994/14.5) = 172 • We need to find the value of: (29.94/1.45) • Convert the target decimals using powers of 10: \frac{2994 × 10^{-2}}{145 × 10^{-2}} Wait, the original denominator is 14.5. Let us adjust the target fraction to match the known format: (29.94/1.45) = (29.94 × 100/1.45 × 10) × (1/10) = (2994/14.5) × (1/10) • Substitute the known value (172): 172 × (1/10) = 17.2 Exam Strategy & Shortcut: Compare the new numbers to the old numbers. The dividend (2994) becomes smaller by a factor of 100. The divisor (14.5) becomes smaller by a factor of 10. The net change to the quotient is division by 10 (100 \div 10 = 10). Thus, 172 \div 10 = 17.2. Common Pitfall: Students often apply the 100 shift and the 10 shift in the wrong direction, mistakenly arriving at 1.72 or 1720. Remember that decreasing the dividend decreases the result, but decreasing the divisor increases the result. Final Answer: Therefore, the correct answer is 17.2.

Question 9

If $\frac{1}{6.198} = 0.16134$, then the value of $\frac{1}{0.0006198}$ is

Accepted Answer

Concept & Logic: Moving the decimal point in the denominator of a fraction results in an inverse shift in the overall value of the fraction. Step-by-Step Solution: • The given relation is: (1/6.198) = 0.16134 • We need to find the value of: (1/0.0006198) • Relate the new denominator to the given denominator: 0.0006198 = 6.198 × 10^{-4} • Substitute this back into the target fraction: (1/6.198 × 10^{-4)} = (1/6.198) × (1/10^{-4)} = 0.16134 × 10^{4} • Multiply by 10000 (shift the decimal 4 places to the right): = 1613.4 Exam Strategy & Shortcut: The denominator decreased by 4 decimal places (from 6.198 to 0.0006198). Making a denominator smaller makes the overall fraction larger. Shift the decimal point of the given answer (0.16134) exactly 4 places to the right to compensate. Common Pitfall: Multiplying the given value by 10^{-4} instead of 10^{4}. Remember that a smaller denominator yields a larger result. Final Answer: Therefore, the correct answer is 1613.4.

Question 10

Given $168 	imes 32 = 5376$, then $5.376 \div 16.8$ is equal to

Accepted Answer

Concept & Logic: Multiplication and division are inverse operations. If A × B = C, then C \div A = B. Decimal points shift this base result by powers of 10. Step-by-Step Solution: • Start with the given base relationship: 168 × 32 = 5376 This can be rewritten as: (5376/168) = 32 • The target expression to evaluate is: (5.376/16.8) • Convert the decimal numbers to their integer equivalents with powers of 10: = \frac{5376 × 10^{-3}}{168 × 10^{-1}} • Group the integers and simplify the powers: = ( (5376/168) ) × 10^{(-3 - (-1))} = 32 × 10^{-2} = 0.32 Exam Strategy & Shortcut: The dividend 5.376 has 3 decimal places, and the divisor 16.8 has 1 decimal place. The quotient will have 3 - 1 = 2 decimal places. Apply this to the base integer answer of 32 to get 0.32. Common Pitfall: Miscounting the net decimal places. If a student adds the decimal places instead of subtracting them during division, they might choose 0.0032 or a similarly incorrect option. Final Answer: Therefore, the correct answer is 0.32.

Question 11

$\frac{5.3472 	imes 324.23}{3.489 	imes 5.42}$ is the same as

Accepted Answer

Concept & Logic: For a complex fraction to maintain its value, the net decimal multiplier (total numerator decimal places minus total denominator decimal places) must remain identical across the original and equivalent expressions. Step-by-Step Solution: • Analyze the decimal places in the original expression: Numerator: 5.3472 (4 places) and 324.23 (2 places) \rightarrow Total 6 places (10^{-6}). Denominator: 3.489 (3 places) and 5.42 (2 places) \rightarrow Total 5 places (10^{-5}). • Calculate the net shift of the original expression: Net multiplier = \frac{10^{-6}}{10^{-5}} = 10^{-1}. This means the raw integer fraction must be multiplied by 10^{-1}. • Test the options to find the one with a net multiplier of 10^{-1}: Option (a): Numerator (0 + 4 = 4), Denominator (3 + 1 = 4). Net = 10^0 = 1. (Incorrect) Option (b): Numerator (0 + 0 = 0), Denominator (0 + 0 = 0). Net = 10^0 = 1. (Incorrect) Option (c): Numerator (2 + 2 = 4), Denominator (2 + 2 = 4). Net = 10^0 = 1. (Incorrect) Option (d): Numerator (0 + 1 = 1), Denominator (0 + 0 = 0). Net = \frac{10^{-1}}{10^0} = 10^{-1}. (Correct) Exam Strategy & Shortcut: Sum the decimal places on top (4+2=6) and on bottom (3+2=5). The top has 1 more decimal place than the bottom. Look at the options and find the one where the top has exactly 1 more decimal place than the bottom. Only Option (d) has 1 decimal place on top and 0 on the bottom. Common Pitfall: Assuming that clearing all decimals (as in Option b) creates an equivalent fraction. Clearing all decimals only works if the original numerator and denominator had the exact same number of total decimal places. Final Answer: Therefore, the correct answer is (53472 × 3242.3/3489 × 542).

Question 12

The sum of the first 20 terms of the series $\frac{1}{5 	imes 6} + \frac{1}{6 	imes 7} + \frac{1}{7 	imes 8} + \dots$ is

Accepted Answer

Concept & Formula: This is a telescoping series where each individual term can be split using partial fractions: (1/n(n+1)) = (1/n) - (1/n+1) Step-by-Step Solution: • Identify the form of the n-th term: T_n = (1/(n+4)(n+5)) • Determine the 20th term of this series: For n = 1, the term is (1/5 × 6). For n = 20, the term is (1/(20+4) × (20+5)) = (1/24 × 25). • Expand the sum of the 20 terms using the splitting method: S_{20} = ((1/5) - (1/6)) + ((1/6) - (1/7)) + \dots + ((1/24) - (1/25)) • Cancel out the intermediate terms (telescoping property): S_{20} = (1/5) - (1/25) • Compute the final fractional and decimal value: S_{20} = (5 - 1/25) = (4/25) (4/25) = 0.16 Exam Strategy & Shortcut: For any standard telescoping series with a difference of 1 in the denominator factors, the sum is always: Sum = First Factor^{-1} - Last Factor^{-1} Here, the first factor is 5, and the last factor after 20 terms is 5 + 20 = 25. (1/5) - (1/25) = (4/25) = 0.16 Common Pitfall: Mistakenly calculating the 20th term's denominator as 20 × 21 instead of tracking the initial offset starting at 5. Always verify the first term match before computing the final term. Final Answer: Therefore, the correct answer is 0.16.

Question 13

The last digit in the decimal representation of $\left(\frac{1}{5}\right)^{2000}$ is

Accepted Answer

Concept & Logic: Convert the fraction into its standard decimal base. The behavior of powers applied to decimals determines the terminating trailing digit. Step-by-Step Solution: • Express the fraction as a decimal: (1/5) = 0.2 • The given term becomes: (0.2)^{2000} = ((2/10))^{2000} = \frac{2^{2000}}{10^{2000}} • Analyze the cyclicity of the last digit of powers of 2: • 2^1 = 2 • 2^2 = 4 • 2^3 = 8 • 2^4 = 16 (ends in 6) The cyclicity length is 4. • Find the remainder of the exponent when divided by 4: 2000 \div 4 = 500 \quad (Remainder 0) A remainder of 0 corresponds to the 4th position in the cycle, meaning 2^{2000} ends exactly in the digit 6. • Dividing an integer ending in 6 by a power of 10 shifts the decimal point but leaves the final non-zero terminating rightmost digit as 6. Exam Strategy & Shortcut: Recognize that ((1/5))^n = (0.2)^n. The last digit matches the unit digit cyclicity pattern of 2^n. Because 2000 is perfectly divisible by 4, the units digit matches 2^4 = 16, which ends in 6. Common Pitfall: Students often confuse "last digit in the decimal representation" with "first decimal place", picking 5 because they think of the fraction 5 dynamically, or selecting 2 without applying the exponent cycle. Final Answer: Therefore, the correct answer is 6.

Question 14

Which of the following is equal to 1?

Accepted Answer

Concept & Logic: For a fraction involving powers of matching base significant digits to equal 1, the total power of 10 in the numerator must balance out perfectly with the total power of 10 in the denominator. Step-by-Step Solution: • Let us verify option (d) by breaking it down using powers of 10 relative to the base integer 11: Numerator = (0.11)^2 = (11 × 10^{-2})^2 = 11^2 × 10^{-4} Denominator = 11^2 × 0.01 = 11^2 × 10^{-2} Let's recheck option (b): Numerator = (1.1)^2 = (11 × 10^{-1})^2 = 11^2 × 10^{-2} Denominator = 11^2 × (0.01)^2 = 11^2 × (10^{-2})^2 = 11^2 × 10^{-4} Let's compute the exact value of option (a): \frac{11^2 × 10^{-4}}{(11 × 10^{-1})^2 × 10^{-1}} = \frac{11^2 × 10^{-4}}{11^2 × 10^{-2} × 10^{-1}} = \frac{10^{-4}}{10^{-3}} = 0.1 Let's recalculate option (d) directly: ((0.11)^2/11^2 × 0.01) = (0.0121/121 × 0.01) = (0.0121/1.21) = 0.01 Let's carefully check option (c): Numerator = (0.011)^2 = (11 × 10^{-3})^2 = 11^2 × 10^{-6} Denominator = (1.1)^2 × (0.01)^2 = (11 × 10^{-1})^2 × (10^{-2})^2 = 11^2 × 10^{-2} × 10^{-4} = 11^2 × 10^{-6} Fraction Value = \frac{11^2 × 10^{-6}}{11^2 × 10^{-6}} = 1 Exam Strategy & Shortcut: Count total decimal places on both sides of each option, squaring shifts appropriately: Option (c): Top has 3 × 2 = 6 decimal places. Bottom has (1 × 2) + (2 × 2) = 2 + 4 = 6 decimal places. Since the integer base elements 11 cancel out completely, matching decimal counts ensure the ratio equals 1. Common Pitfall: Miscounting decimal counts during squaring (e.g., assuming (0.01)^2 has 4 decimal places but adding them incorrectly to other indices). Final Answer: Therefore, the correct answer is ((0.011)^2/(1.1)^2 × (0.01)^2).

Question 15

If $1.5x = 0.04y$, then the value of $\left(\frac{y - x}{y + x}\right)$ is

Accepted Answer

Concept & Strategy: Find the ratio of x to y from the initial linear equation, then substitute those fractional values directly into the target rational expression. Step-by-Step Solution: • Rearrange the given equation to isolate the ratio (x/y): 1.5x = 0.04y \implies (x/y) = (0.04/1.5) • Eliminate the decimals by multiplying the numerator and denominator by 100: (x/y) = (4/150) = (2/75) • We can assign proportional values: x = 2 and y = 75. • Substitute these values into the required target expression: (y - x/y + x) = (75 - 2/75 + 2) = (73/77) Exam Strategy & Shortcut: Use componendo and dividendo logic rules. Once you establish (y/x) = (75/2), the expression (y-x/y+x) is simply (75-2/75+2) = (73/77). This takes less than 15 seconds. Common Pitfall: Accidentally reversing the ratio values as (x/y) = (75/2), which leads to a sign error or choosing an incorrect distractor value. Final Answer: Therefore, the correct answer is (73/77).

Question 16

The value of $\left[35.7 - \left(3 + \frac{1}{3 + \frac{1}{3}}\right) - \left(2 + \frac{1}{2 + \frac{1}{2}}\right)\right]$ is

Accepted Answer

Concept & Strategy: Solve the nested continued fractions from the bottom upward before carrying out the final decimal subtraction. Step-by-Step Solution: • Evaluate the first continued fraction block: 3 + (1/3) = (10/3) 3 + (1/\frac{10){3}} = 3 + (3/10) = (33/10) = 3.3 • Evaluate the second continued fraction block: 2 + (1/2) = (5/2) 2 + (1/\frac{5){2}} = 2 + (2/5) = (12/5) = 2.4 • Substitute these clean values back into the primary expression: 35.7 - 3.3 - 2.4 = 32.4 - 2.4 = 30 Exam Strategy & Shortcut: Convert fractions to explicit decimals immediately when denominators are friendly (like 10 and 5). Recognizing that (3/10) = 0.3 and (2/5) = 0.4 turns a messy expression into simple arithmetic: 35.7 - 3.3 - 2.4 = 30. Common Pitfall: Forgetting to invert the denominator fraction when resolving the nested layer (e.g., adding (10/3) directly instead of converting it to (3/10)). Final Answer: Therefore, the correct answer is 30.

Question 17

The expression $\frac{(0.1667)(0.8333)(0.3333)}{(0.2222)(0.6667)(0.1250)}$ is approximately equal to

Accepted Answer

Concept & Strategy: Recognize standard recurring decimal fractions and convert them to their exact rational fractional forms (p/q) to simplify the entire product instantly. Step-by-Step Solution: • Convert each decimal value to its exact fraction equivalent: • 0.1667 \approx (1/6) • 0.8333 \approx (5/6) • 0.3333 \approx (1/3) • 0.2222 \approx (2/9) • 0.6667 \approx (2/3) • 0.1250 = (1/8) • Rewrite the complete expression using these fractions: Numerator = (1/6) × (5/6) × (1/3) = (5/108) Denominator = (2/9) × (2/3) × (1/8) = (4/216) = (1/54) • Divide the numerator by the denominator: Value = (5/108) \div (1/54) = (5/108) × 54 = (5/2) = 2.50 Exam Strategy & Shortcut: Cancel matching items early in fractional format: (\frac{1/6) × (5/6) × (1/3)}{(2/9) × (2/3) × (1/8)} The (1/3) terms cancel out. Notice (2/9) × (1/8) = (1/36), and (1/6) × (1/6) = (1/36). These cancel completely, leaving just (5/2 × 2) forms or straightforward evaluation yielding 2.50. Common Pitfall: Attempting long decimal multiplication or imprecise rounding approximations, which leads to floating errors near 2.40 or 2.43. Final Answer: Therefore, the correct answer is 2.50.

Question 18

$$ \frac{1.6 	imes 3.2}{0.08} = x $$

Accepted Answer

Concept & Formula: To clear decimals from a fraction, count the total decimal places in the numerator and the denominator. Multiply both by a power of 10 that eliminates the decimals completely, converting the problem into simple integer arithmetic. Step-by-Step Solution: Given: (1.6 × 3.2/0.08) Calculation: • Count the decimal places: • Numerator: 1.6 (1 place) × 3.2 (1 place) = 2 total decimal places. • Denominator: 0.08 = 2 total decimal places. • Since the number of decimal places in the numerator perfectly matches the denominator (2 places each), the decimals simply cancel each other out. • Rewrite the expression with whole numbers: (16 × 32/8) • Simplify by dividing 32 by 8: (32/8) = 4 • Multiply the remaining terms: 16 × 4 = 64 Exam Strategy & Shortcut: Shift & Cancel Method: Mental math is faster when you shift the decimal. You can shift the decimal of 0.08 two places to the right to make it 8. To balance this, shift the decimal of 1.6 one place (to 16) and 3.2 one place (to 32). The equation immediately becomes 16 × 32 / 8. Recognize that 32 / 8 = 4. Then 16 × 4 = 64. Common Pitfall: A very common mistake is miscounting the decimal places and incorrectly assuming the expression simplifies to 16 × 32 / 80 or similar. Students often see 0.08 and 1.6 × 3.2 and think there are more decimal places in the denominator, leading to an answer off by a factor of 10 (like 6.4). Always sum the decimal places in the numerator before comparing. Final Answer: Therefore, the correct answer is 64.

Question 19

$$ \frac{0.0203 	imes 2.92}{0.0073 	imes 14.5 	imes 0.7} = x $$

Accepted Answer

Concept & Formula: Eliminate decimals by equating the total number of decimal places in the numerator and the denominator. Once converted to integers, rely on prime factorization and spotting multiples to simplify the expression rapidly. Step-by-Step Solution: Given: (0.0203 × 2.92/0.0073 × 14.5 × 0.7) Calculation: • Count the decimal places in the numerator: 0.0203 (4 places) + 2.92 (2 places) = 6 total decimal places. • Count the decimal places in the denominator: 0.0073 (4 places) + 14.5 (1 place) + 0.7 (1 place) = 6 total decimal places. • The decimal places are perfectly balanced (6 on top, 6 on bottom). Remove the decimals completely. • Rewrite as whole numbers: (203 × 292/73 × 145 × 7) • Simplify by dividing numbers that are obvious multiples: 1. Divide 203 by 7: 203 / 7 = 29 2. Look at 292 and 73. Notice that 70 × 4 = 280 and 3 × 4 = 12, so: 292 / 73 = 4 • Substitute these simplified values back into the expression: (29 × 4/145) • Notice the relationship between 29 and 145. Since 30 × 5 = 150, then 29 × 5 is 145. (145/29) = 5 • This leaves 4 in the numerator and 5 in the denominator: (4/5) • Convert the fraction to a decimal: (4/5) = 0.8 Exam Strategy & Shortcut: Estimation and Unit Digits: After stripping the decimals, you have (203 × 292) / (73 × 145 × 7). If you quickly estimate: (200 × 300) / (70 × 150 × 7) \approx 60000 / 70000 \approx 6/7, which is slightly less than 1. Looking at the options, 0.8 is the only logical choice that is slightly less than 1 (1.45, 2.40, and 3.25 are far too large). Common Pitfall: A major trap is trying to multiply out the decimals as they are, like 0.0203 × 2.92 = 0.059276. This takes immense time and almost guarantees an arithmetic mistake. Always balance and drop decimals first. Final Answer: Therefore, the correct answer is 0.8.

Question 20

The value of   $$ \frac{489.1375 	imes 0.0483 	imes 1.956}{0.0873 	imes 92.581 	imes 99.749} $$   is closest to

Accepted Answer

Strategy & Logic: Just like the previous problem, the phrase "closest to" combined with options that differ strictly by a decimal place magnitude (0.006, 0.06, 0.6, 6) signals that you should use Aggressive Approximation. We need to find the order of magnitude, not the exact digit. Step-by-Step Solution: Given: (489.1375 × 0.0483 × 1.956/0.0873 × 92.581 × 99.749) Deduction (Approximation): • Round each term to the nearest easy-to-use whole number or clean decimal: • 489.1375 \approx 500 (or 490) • 0.0483 \approx 0.05 • 1.956 \approx 2 • 0.0873 \approx 0.09 (or 0.1) • 92.581 \approx 90 (or 100) • 99.749 \approx 100 • Let's use these rounded values to rewrite the expression: (500 × 0.05 × 2/0.09 × 90 × 100) • Calculate the approximate numerator: 500 × 0.05 = 25 25 × 2 = 50 • Calculate the approximate denominator: 0.09 × 90 = 8.1 8.1 × 100 = 810 • Evaluate the approximate ratio: (50/810) = (5/81) • Since 5/80 would be exactly 1/16, and we know 1/16 = 0.0625, the value 5/81 is very close to 0.06. • Checking the options, 0.06 perfectly matches our approximated order of magnitude. Exam Strategy & Shortcut: Magnitude Tracking: You can simplify the order of magnitude visually. Numerator is roughly: 500 × (5 × 10^{-2}) × 2 = 50. Denominator is roughly: (1 × 10^{-1}) × 100 × 100 = 1000. 50 / 1000 = 0.05. The closest option to 0.05 is 0.06. This confirms that the magnitude is in the hundredths place, instantly eliminating 0.006, 0.6, and 6. Common Pitfall: Rounding down too many terms in the denominator (e.g., making it 0.08 × 90 × 90) while rounding up the numerator. This can skew your final fraction slightly. Always try to round one term up and another down to balance the error, though in magnitude-based questions, minor skewing rarely changes the final answer. Final Answer: Therefore, the correct answer is 0.06.

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