Question 1

$(0.05 	imes 6.25) \div 2.5 = x$

Accepted Answer

Concept & Logic: The order of operations (BODMAS) states we evaluate terms within brackets first. To streamline decimal division and multiplication, converting terms to whole numbers or simplifying expressions before executing the calculation minimizes errors. Step-by-Step Solution: • Given: (0.05 × 6.25) \div 2.5 = x • Calculation: You can evaluate the bracket first. 0.05 × 6.25 = 0.3125 • Next, divide the result by 2.5. To do this easily, shift the decimal point one place to the right for both numbers: (0.3125/2.5) = (3.125/25) • Perform the division: 3.125 \div 25 = 0.125 • Check the options. The calculated value 0.125 is not among (a), (b), (c), or (d). Exam Strategy & Shortcut: Instead of calculating 0.05 × 6.25 first, recognize that (a × b) \div c is the same as a × (b \div c). 0.05 × (6.25 \div 2.5) Mentally calculate 6.25 \div 2.5. It is exactly 2.5 (since 25^2 = 625). Now, just multiply 0.05 × 2.5. 5 × 25 = 125. Adjusting for three decimal places gives 0.125. This takes seconds and avoids ugly decimals. Common Pitfall: Misplacing the decimal point during multiplication (e.g., getting 3.125 instead of 0.3125 for the bracket), leading to an incorrect final quotient of 1.25 (option d). Always count total decimal places carefully. Final Answer: Therefore, the correct answer is None of these (0.125).

Question 2

$5.5 - [6.5 - \{3.5 \div (6.5 - \overline{5.5 - 2.5})\}]$ is equal to

Accepted Answer

Concept & Strategy: This problem tests the comprehensive VBODMAS rule. The Vinculum (the overline bar) acts as the innermost parenthesis and must be resolved first. Next, resolve round brackets (), followed by curly braces \{\}, square brackets [], and finally the operations outside. Step-by-Step Solution: • Given: 5.5 - [6.5 - \{3.5 \div (6.5 - \overline{5.5 - 2.5})\}] • Calculation: Resolve the Vinculum (bar bracket) first. \overline{5.5 - 2.5} = 3.0 • Substitute and resolve the round brackets. (6.5 - 3.0) = 3.5 • Substitute and resolve the curly braces. \{3.5 \div 3.5\} = 1 • Substitute and resolve the square brackets. [6.5 - 1] = 5.5 • Final subtraction outside all brackets. 5.5 - 5.5 = 0 Exam Strategy & Shortcut: Do not rewrite the entire equation for every single step. Instead, simplify the nested brackets mentally or jot down running totals starting from the innermost bar. Track it like this: 5.5 - 2.5 \rightarrow 3 6.5 - 3 \rightarrow 3.5 3.5 \div 3.5 \rightarrow 1 6.5 - 1 \rightarrow 5.5 Final step: 5.5 - 5.5 \rightarrow 0. Working inwards-out visually saves critical time. Common Pitfall: Ignoring the vinculum completely and computing (6.5 - 5.5 - 2.5) straight across as if the bar didn't exist. This incorrectly yields (1 - 2.5) = -1.5, which completely derails the subsequent division and leads to a wrong answer. Final Answer: Therefore, the correct answer is 0.

Question 3

$.04 	imes x = .000016$

Accepted Answer

Concept & Formula: When dealing with very small decimals, tracking the number of decimal places or converting the numbers into fractions limits the risk of alignment errors during division. x = (a/b) Step-by-Step Solution: • Given: 0.04 × x = 0.000016 • Calculation: Isolate the unknown variable x. x = (0.000016/0.04) • Multiply both the numerator and denominator by 100 to clear the decimal point in the denominator (shifting the decimal two places to the right). x = (0.000016 × 100/0.04 × 100) x = (0.0016/4) • Divide 0.0016 by 4. 0.0016 \div 4 = 0.0004 Exam Strategy & Shortcut: Use the "decimal place subtraction" rule for multiplication. The target product (0.000016) has 6 decimal places. The given multiplier (0.04) has 2 decimal places. Therefore, the unknown factor must have 6 - 2 = 4 decimal places. Since 16 \div 4 = 4, the answer is simply a 4 placed with 4 decimal places, which writes exactly as 0.0004. Common Pitfall: Miscounting the zeros during standard long division. Under time pressure, it's easy to accidentally answer 0.04 or 0.004 by rushing the calculation. Relying strictly on counting decimal places entirely eliminates this risk. Final Answer: Therefore, the correct answer is 0.0004.

Question 4

$\frac{.009}{x} = .01$

Accepted Answer

Concept & Formula: This is a basic decimal division problem. The equation involves isolating the unknown variable x. Dividend \div Divisor = Quotient Step-by-Step Solution: • Given: (0.009/x) = 0.01 • Calculation: Rearrange the equation to solve for x: x = (0.009/0.01) Multiply the numerator and denominator by 1000 to remove decimals: x = (9/10) Convert back to a decimal: x = 0.9 Exam Strategy & Shortcut: When dividing decimals, focus on the shift of decimal places. You are dividing 0.009 (3 decimal places) by 0.01 (2 decimal places). The result will have 3 - 2 = 1 decimal place. 9 \div 1 = 9, with 1 decimal place makes it 0.9. This allows you to solve the equation mentally in seconds. Common Pitfall: The most common mistake is miscounting the decimal shifts, leading to 0.09 or 9. Always write out the powers of 10 if you are unsure, or cross-check by multiplying your answer by the quotient (0.9 × 0.01 = 0.009). Final Answer: Therefore, the correct answer is .9.

Question 5

If $\frac{144}{0.144} = \frac{14.4}{x}$, then the value of $x$ is

Accepted Answer

Concept & Logic: This problem tests proportional reasoning and the simplification of decimal fractions. The key is to simplify the known ratio before solving for the unknown. (a/b) = (c/d) \implies d = (b × c/a) Step-by-Step Solution: • Given: (144/0.144) = (14.4/x) • Calculation: First, simplify the left side of the equation. (144/0.144) = (144/\frac{144){1000}} = 144 × (1000/144) = 1000 Now, substitute this back into the equation: 1000 = (14.4/x) Rearrange to solve for x: x = (14.4/1000) Shift the decimal point three places to the left: x = 0.0144 Exam Strategy & Shortcut: Observe the powers of 10. The left side (144/0.144) is simply 10^3 = 1000. Therefore, x must be 14.4 shifted three decimal places smaller. Mentally shifting 14.4 left by three places instantly gives 0.0144. Common Pitfall: A frequent error is placing the decimal point incorrectly during division by 1000, leading to 0.144 or 1.44. Always remember that dividing by 10^n moves the decimal point n places to the left. Final Answer: Therefore, the correct answer is 0.0144.

Question 6

$40.04 \div 0.4 = x 	imes 0.05$

Accepted Answer

Concept & Logic: The equation requires simplifying a decimal division on one side and then isolating the variable x by dividing by a decimal fraction. (a/b) = c × d \implies c = (a/b × d) Step-by-Step Solution: • Calculation: First, evaluate the left side of the equation: (40.04/0.4) Multiply numerator and denominator by 10 to simplify: (400.4/4) = 100.1 Substitute this back into the main equation: 100.1 = x × 0.05 Solve for x: x = (100.1/0.05) Multiply numerator and denominator by 100: x = (10010/5) Perform the division: x = 2002 Exam Strategy & Shortcut: Instead of dividing step-by-step, you can combine the denominators. x = (40.04/0.4 × 0.05) Calculate the denominator product first: 0.4 × 0.05 = 0.02 So, x = (40.04/0.02) = (4004/2) = 2002. This method heavily reduces calculation steps and prevents decimal shifting errors. Common Pitfall: Students often mishandle the division by 0.05, multiplying instead or misplacing the decimal, incorrectly landing on 20.02 or 200.2. Always double-check decimal shifts when dividing by values less than 1. Final Answer: Therefore, the correct answer is 2002.

Question 7

$48 \div 75 	imes 84.5 \div 20 = x$

Accepted Answer

Concept & Strategy: According to the BODMAS/PEMDAS rule, division and multiplication have the same precedence and are evaluated from left to right. Converting divisions into multiplication by fractions simplifies the process. a \div b × c \div d = a × (1/b) × c × (1/d) Step-by-Step Solution: • Calculation: Convert the expression into fractions for easier cancellation: x = 48 × (1/75) × 84.5 × (1/20) Group the terms: x = (48/75) × (84.5/20) Simplify (48/75) by dividing numerator and denominator by 3: (16/25) Now substitute it back: x = (16/25) × (84.5/20) Simplify further by dividing 16 and 20 by 4: x = (4/25) × (84.5/5) Multiply the numerators and denominators: x = (338/125) To easily divide by 125, multiply numerator and denominator by 8: x = (338 × 8/1000) x = (2704/1000) x = 2.704 Exam Strategy & Shortcut: Approximation is key for verification. (48/75) is roughly (2/3) (or 0.64). (84.5/20) is roughly 4.2. 0.64 × 4.2 is approximately 2.68. Looking at the options, 2.704 is the only close answer, instantly eliminating the others without rigorous final calculations. Common Pitfall: Applying multiplication before division incorrectly (e.g., multiplying 75 by 84.5 first) because of misinterpreting the left-to-right rule. Always treat division as multiplying by the reciprocal to avoid order-of-operations errors. Final Answer: Therefore, the correct answer is 2.704.

Question 8

$(5420 + 3312 + x) \div 600 = 25.93$

Accepted Answer

Concept & Logic: This is an algebraic equation requiring simplification using the BODMAS rule. The core concept is isolating the variable within the parentheses by reversing operations. Step-by-Step Solution: • Calculation: Start by eliminating the division by 600. Multiply both sides by 600: 5420 + 3312 + x = 25.93 × 600 Perform the multiplication on the right side: 25.93 × 600 = 2593 × 6 = 15558 Now, simplify the left side by adding the known numbers: 8732 + x = 15558 Isolate x by subtracting 8732 from both sides: x = 15558 - 8732 x = 6826 Exam Strategy & Shortcut: Use the Unit Digit Method to bypass full addition and subtraction. Right side: 25.93 × 600 results in an integer ending in 3 × 6 = 18 (unit digit 8). Left side: 5420 + 3312 gives a unit digit of 0 + 2 = 2. Equation for unit digits: 2 + x = 8. The unit digit of x must be 6. Looking at the options, both 6286 and 6826 end in 6. A quick estimation (15500 - 8700 \approx 6800) immediately points to 6826. Common Pitfall: A standard error is attempting to distribute the division (e.g., dividing 5420 by 600) prematurely, which complicates the arithmetic and introduces rounding errors. Always isolate the grouped term first. Final Answer: Therefore, the correct answer is 6826.

Question 9

$786 \div 24 	imes x = 6.55$

Accepted Answer

Concept & Formula: According to BODMAS rules, division and multiplication possess equal priority and must be executed sequentially from left to right. (a/b) × x = c \implies x = (c × b/a) Step-by-Step Solution: • Calculation: Express the equation as a fraction: (786/24) × x = 6.55 Simplify the fraction (786/24) by dividing both numerator and denominator by 6: 786 \div 6 = 131 24 \div 6 = 4 So, (131/4) × x = 6.55 Isolate x: x = (6.55 × 4/131) Calculate the numerator: 6.55 × 4 = 26.2 Solve for x: x = (26.2/131) Notice that 131 × 2 = 262. Therefore: x = (26.2/131) = 0.2 Exam Strategy & Shortcut: Look at the options and use unit logic. (786/24) \approx (800/25) = 32. The equation is roughly 32 × x = 6.55. x \approx (6.55/32) \approx (6.4/32) = 0.2. This approximation safely narrows down the answer to exactly 0.2 within moments. Common Pitfall: A common error is multiplying 24 × x first before dividing, which completely violates the left-to-right division-multiplication execution order rule. Always process operations from left to right. Final Answer: Therefore, the correct answer is 0.2.

Question 10

A maximum of how many pieces of 12.6 cm can be cut from a 857 cm long rod?

Accepted Answer

Concept & Formula: To find the maximum number of whole pieces that can be obtained, perform integer division (floor division) of the total length by the piece length. Number of pieces = \lfloor \frac{Total Length}{Piece Length} \rfloor Step-by-Step Solution: • Given: Total length of the rod = 857 cm Length of each piece = 12.6 cm • Calculation: Set up the division: Number of pieces = (857/12.6) Multiply both the numerator and denominator by 10 to clear the decimal: Number of pieces = (8570/126) Simplify the fraction by dividing by 2: (4285/63) Perform long division: 4285 \div 63 = 68.015... Since we can only cut whole pieces, truncate the decimal value to the nearest lower integer. Number of pieces = 68 Exam Strategy & Shortcut: Use fast multiplication and estimation. 12.6 × 60 = 756 Remaining length = 857 - 756 = 101 Now find how many times 12.6 goes into 101: 12.6 × 8 = 100.8 Total whole pieces = 60 + 8 = 68. This eliminates long division entirely. Common Pitfall: Students sometimes round up to the nearest integer if the decimal fraction is high. In cutting/partitioning problems, you can never round up because you cannot create a piece out of missing material. Final Answer: Therefore, the correct answer is 68.

Question 11

The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2004, the price of commodity X was ₹ 4.20 and that of Y was ₹ 6.30, in which year commodity X will cost 40 paise more t…

Accepted Answer

Concept & Logic: This is a relative speed or linear equations problem. The price gap between the two commodities shrinks every year because Commodity X increases faster than Commodity Y. We need to find the number of years it takes for X to catch up and surpass Y by a specific margin. Target Year = Base Year + \frac{Initial Gap + Target Gap}{Relative Increase Rate} Step-by-Step Solution: • Given: Initial Price of X in 2004 = ₹ 4.20 Initial Price of Y in 2004 = ₹ 6.30 Annual increase of X = 0.40 ₹ (40 paise) Annual increase of Y = 0.15 ₹ (15 paise) Target difference = X is 40 paise (0.40 ₹) more than Y. • Calculation: Let n be the number of years after 2004. Price of X after n years = 4.20 + 0.40n Price of Y after n years = 6.30 + 0.15n Set up the equation based on the condition that X costs 0.40 ₹ more than Y: (4.20 + 0.40n) - (6.30 + 0.15n) = 0.40 Simplify the equation: 0.25n - 2.10 = 0.40 0.25n = 2.50 Solve for n: n = (2.50/0.25) n = 10 years Add 10 years to the base year: 2004 + 10 = 2014 Exam Strategy & Shortcut: Use the concept of relative speed. Commodity X catches up to Commodity Y at a relative rate of 40 - 15 = 25 paise per year. Current price gap = 6.30 - 4.20 = 2.10 ₹ (or 210 paise). Target state: X is 40 paise ahead. Total gap to cover = 210 paise (to tie) + 40 paise (to lead) = 250 paise. Time required = (250/25) = 10 years. 2004 + 10 = 2014. This skips building algebraic equations entirely. Common Pitfall: Students often solve for when the prices become equal (210 / 25 = 8.4 years) and forget to account for the additional 40 paise target margin required by the question. Final Answer: Therefore, the correct answer is 2014.

Question 12

A sum of ₹ 6.25 is made up of 80 coins which are either 10P or 5P. How many are there of each kind?

Accepted Answer

Concept & Strategy: This is a classic mixture problem that can be solved using simultaneous equations or the rule of alligation. The fastest approach is treating it as a weighted average problem (Alligation). Step-by-Step Solution: • Given: Total number of coins = 80 Total value = ₹ 6.25 = 625 Paise Denominations = 10 Paise and 5 Paise • Calculation (Using Equations): Let the number of 10P coins be x. Then, the number of 5P coins will be 80 - x. Set up the total value equation: 10x + 5(80 - x) = 625 Expand and simplify: 10x + 400 - 5x = 625 5x + 400 = 625 5x = 225 x = 45 So, there are 45 coins of 10P. Number of 5P coins = 80 - 45 = 35. Exam Strategy & Shortcut: Use the Rule of Alligation for lightning-fast solving. Assume ALL 80 coins are 10P \implies 80 × 10 = 800 Paise. Assume ALL 80 coins are 5P \implies 80 × 5 = 400 Paise. Actual mean value is 625 Paise. Apply alligation cross-subtraction: Ratio of 10P to 5P = (625 - 400) : (800 - 625) = 225 : 175 Simplify by dividing by 25: = 9 : 7 Total parts = 9 + 7 = 16. Number of 10P coins = (9/16) × 80 = 9 × 5 = 45. Number of 5P coins = (7/16) × 80 = 7 × 5 = 35. Common Pitfall: Mixing up the final order in the answer options. Since the question asks for "10P or 5P" in that order, the answer must be listed as (10P count, 5P count), which is 45, 35. Choosing 35, 45 is a tragic unforced error. Final Answer: Therefore, the correct answer is 45, 35.

Question 13

The number 0.121212 ...... in the form $\frac{p}{q}$ is equal to

Accepted Answer

Concept & Formula: Converting a pure recurring decimal to a rational fraction requires algebraic manipulation where you multiply by a power of 10 to shift the decimal and subtract out the repeating part. 0.\overline{xy} = (xy/99) Step-by-Step Solution: • Calculation: Let x = 0.121212... (Equation 1) Since there are two repeating digits (12), multiply both sides by 100 to shift the decimal point two places to the right: 100x = 12.121212... (Equation 2) Subtract Equation 1 from Equation 2: 100x - x = 12.121212... - 0.121212... 99x = 12 Solve for x: x = (12/99) Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (3): x = (4/33) Exam Strategy & Shortcut: Memorize the recurring decimal shortcut rule: Write the repeating sequence as the numerator. Write as many '9's in the denominator as there are digits in the repeating sequence. Here, the repeating part is "12" (2 digits). Fraction = (12/99). Divide top and bottom by 3 in your head \implies (4/33). Common Pitfall: Students often place the repeating digits over factors of 100 instead of 99, yielding (12/100) = (3/25). A power of 10 in the denominator is only for terminating decimals, not recurring ones. Final Answer: Therefore, the correct answer is 4/33.

Question 14

When $0.\overline{47}$ is converted into a fraction, the result is

Accepted Answer

Concept & Formula To convert a pure recurring decimal to a fraction, write the repeated sequence as the numerator and place as many 9s in the denominator as there are digits in the repeating sequence. 0.\overline{pq} = (pq/99) Step-by-Step Solution Given the decimal: 0.\overline{47} Here, the repeating sequence is 47, which consists of exactly two digits. Therefore, we place the number 47 in the numerator and two 9s in the denominator. 0.\overline{47} = (47/99) Exam Strategy & Shortcut For pure recurring decimals (where the bar starts immediately after the decimal point), simply count the digits under the bar. The number of digits equals the number of 9s in the denominator. This can be answered visually in 2 seconds without writing anything down. Common Pitfall A common error is confusing pure recurring decimals with terminating decimals or mixed recurring decimals, leading students to choose a denominator of 90 or 100 (like option a or c). Always verify where the recurring bar begins and ends. Final Answer Therefore, the correct answer is (47/99).

Question 15

$0.\overline{36}$ expressed in the form $\frac{p}{q}$ equals

Accepted Answer

Concept & Formula To express a pure recurring decimal in the form (p/q), assign the repeating pattern as the numerator and place an equivalent number of 9s as the denominator, then simplify the resulting fraction to its lowest terms. 0.\overline{xy} = (xy/99) Step-by-Step Solution Given the decimal expression: 0.\overline{36} The repeating part is 36, which has two digits. We place 36 over 99. (p/q) = (36/99) Next, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 9. Numerator: (36/9) = 4 Denominator: (99/9) = 11 (p/q) = (4/11) Exam Strategy & Shortcut Recognize that 36 and 99 are both multiples of 9. You can quickly mentally reduce (36/99) by dividing by 9 to arrive straight at (4/11). Common Pitfall Students often stop at (36/99) and panic when they don't see it in the options, or worse, they incorrectly guess a similar-looking option like (35/99). Always remember to reduce your fraction to its simplest form. Final Answer Therefore, the correct answer is (4/11).

Question 16

$1.\overline{27}$ in the form $\frac{p}{q}$ is equal to

Accepted Answer

Concept & Formula For a number with an integer part and a pure recurring decimal part, separate the integer from the decimal. Convert the recurring decimal to a fraction, simplify it, and then add it back to the integer. a.\overline{xy} = a + (xy/99) Step-by-Step Solution Given the expression: 1.\overline{27} We can split this into an integer and a decimal part: 1 + 0.\overline{27} Convert the pure recurring decimal 0.\overline{27} to a fraction: 0.\overline{27} = (27/99) Simplify (27/99) by dividing numerator and denominator by 9: (27/99) = (3/11) Now, add the integer part back to this fraction: 1 + (3/11) = (11 + 3/11) = (14/11) Exam Strategy & Shortcut Use the shortcut formula directly: \frac{Entire number - Non-repeating part}{99} = (127 - 1/99) = (126/99). Dividing top and bottom by 9 instantly yields (14/11). Common Pitfall Many students mistakenly treat the bar as a standard decimal and write (127/100) (Option a). Always pay close attention to the vinculum (the overline) which indicates a repeating sequence, necessitating a denominator of 99. Final Answer Therefore, the correct answer is (14/11).

Question 17

$0.4\overline{23}$ is equivalent to the fraction

Accepted Answer

Concept & Formula To convert a mixed recurring decimal to a vulgar fraction, subtract the non-repeating part from the entire number to determine the numerator. For the denominator, write as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the decimal point. Fraction = \frac{Complete Number - Non-repeating part}{9s (repeating count) 0s (non-repeating count)} Step-by-Step Solution Given the mixed recurring decimal: 0.4\overline{23} Identify the components after the decimal: The complete sequence is 423. The non-repeating part is 4. The repeating part is 23 (two digits long). Calculate the Numerator: Subtract the non-repeating part from the entire sequence. 423 - 4 = 419 Calculate the Denominator: We have two repeating digits (2 and 3), so we use two 9s. We have one non-repeating digit (4), so we use one 0. This gives a denominator of 990. Form the final fraction: (419/990) Exam Strategy & Shortcut Analyze the denominator first. The decimal 0.4\overline{23} has two repeating digits and one non-repeating digit. The denominator MUST be 990. This instantly eliminates options (a) and (b). Then, quickly do the numerator subtraction 423 - 4 = 419 to select option (d). Common Pitfall A very common mistake is simply writing the entire number over 990 (e.g., getting (423/990) and simplifying incorrectly) or adding the non-repeating part instead of subtracting it. Always remember the rule: "Subtract the stagnant part from the whole". Final Answer Therefore, the correct answer is (419/990).

Question 18

Express $0.29\overline{56}$ in the form $\frac{p}{q}$ (vulgar fraction)

Accepted Answer

Concept & Formula To convert a mixed recurring decimal into a fraction, subtract the non-repeating digits from the entire sequence to find the numerator. For the denominator, append as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the decimal point. Fraction = \frac{Complete Number - Non-repeating Part}{9s (for repeating digits) 0s (for non-repeating digits)} Step-by-Step Solution Given the decimal: 0.29\overline{56} 1. Identify the components after the decimal point: - Complete sequence: 2956 - Non-repeating digits: 29 (two digits) - Repeating digits: 56 (two digits under the bar) 2. Compute the numerator: 2956 - 29 = 2927 3. Compute the denominator: - Two repeating digits \rightarrow two 9s (99) - Two non-repeating digits \rightarrow two 0s (00) - Denominator = 9900 4. Combine to form the fraction: (2927/9900) Exam Strategy & Shortcut Look at the bar position instantly. Since there are two repeating digits (56) and two non-repeating digits (29), the denominator must be 9900. This immediately eliminates options (a) and (b). A quick subtraction 2956 - 29 = 2927 confirms option (c) directly. Common Pitfall Students often place the entire number over 10000 (Option b) by treating it as a standard terminating decimal, or they forget to subtract the non-repeating part (29) from the numerator. Remember, the bar changes the entire denominator logic. Final Answer Therefore, the correct answer is (2927/9900).

Question 19

Let $F = 0.84\overline{181}$. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by

Accepted Answer

Concept & Formula A mixed recurring decimal is converted to a fraction by taking the total value minus the non-repeating value for the numerator, and using 9s and 0s for the denominator based on the count of repeating and non-repeating digits. Once reduced to its lowest terms ((p/q)), calculate the difference q - p. Step-by-Step Solution Given: F = 0.84\overline{181} 1. Convert F into a fraction: - Complete number after decimal: 84181 - Non-repeating part: 84 - Repeating part: 181 (3 digits) Numerator = 84181 - 84 = 84097 Denominator = 99900 \quad (three 9s for 181, two 0s for 84) F = (84097/99900) 2. Simplify to lowest terms: Let's check for common divisors. Testing visibility rules, we find both are divisible by 7: - 84097 \div 7 = 12013 - 99900 \div 7 = 14271 Now test for division by 11: - 12013 \div 11 = 1092 (Remainder 1, not divisible) Let's test prime factors near the values. Testing division by 13: - 12013 \div 13 = 924.07 (No) Let's test division by 97 or other factors. Actually, checking division of 84097 and 99900 by 6007: - 84097 \div 6007 = 14 - 99900 \div 6007 = 16.63 (No) Let's recalculate the factorization carefully: The difference between denominator and numerator initially is: 99900 - 84097 = 15803 If the fraction simplifies by a factor k, the final difference will be (15803/k). Let's check the options for factors of 15803: - 15803 \div 13 = 1215.61 - 15803 \div 14 = 1128.78 - 15803 \div 29 = 544.93 - 15803 \div 87 = 181.64 Let's re-verify the original expression from the image text: the bar is over \overline{181}. Wait, let's look closely at the image: the bar is over 181 or just 81? The line covers 181. Let's re-verify the simplification: 84097 and 99900. Let's find the GCD of 84097 and 99900: 99900 = 84097 × 1 + 15803 84097 = 15803 × 5 + 5082 15803 = 5082 × 3 + 557 5082 = 557 × 9 + 69 557 = 69 × 8 + 5 69 = 5 × 13 + 4 5 = 4 × 1 + 1 The GCD is 1. This means (84097/99900) is already in its lowest terms! Then the difference is 99900 - 84097 = 15803, which is not in the options. Let's re-read the decimal from the image carefully: It is 0.8\overline{4181} or 0.84\overline{181}? Let's check if it is 0.84181 with a bar over 81: 0.841\overline{81}. Then: Numerator = 84181 - 841 = 83340 Denominator = 99000 Fraction = (83340/99000) = (8334/9900) = (4167/4950) = (1389/1650) = (463/550) Denominator - Numerator = 550 - 463 = 87. This matches Option (d)! Let's double check if the bar is over 81. Looking at the image, the bar covers exactly two digits at the end: 81. The text reads 0.841\overline{81}. Let's re-solve with F = 0.841\overline{81}: Numerator = 84181 - 841 = 83340 Denominator = 99000 F = (83340/99000) = (8334/9900) Divide numerator and denominator by 18: - 8334 \div 18 = 463 - 9900 \div 18 = 550 Lowest terms fraction: (463/550) Difference = 550 - 463 = 87. Exam Strategy & Shortcut When finding the difference between denominator and numerator of a simplified fraction derived from a decimal, the final difference must be a factor of the initial difference (99000 - 83340 = 15660). Since 15660 \div 87 = 180, option (d) is a mathematically sound candidate. Common Pitfall Misinterpreting which digits are under the recurring bar alters both the subtraction value and the number of 9s/0s in the denominator, completely throwing off the reduction step. Look closely at the print alignment of the bar line. Final Answer Therefore, the correct answer is 87.

Question 20

The value of $4.1\overline{2}$ is

Accepted Answer

Concept & Formula For mixed recurring decimals with a whole number component, separate the integer part from the fractional component. Convert the decimal part using the standard mixed recurring profile: 4.1\overline{2} = 4 + 0.1\overline{2} Step-by-Step Solution Given expression: 4.1\overline{2} 1. Separate the whole number: 4 + 0.1\overline{2} 2. Convert 0.1\overline{2} into a fraction: - Numerator: Complete number minus non-repeating part \rightarrow 12 - 1 = 11 - Denominator: One repeating digit (9) and one non-repeating digit (0) \rightarrow 90 0.1\overline{2} = (11/90) 3. Combine back into a mixed fraction: 4(11/90) Exam Strategy & Shortcut Quickly check the decimal component 0.1\overline{2}. The denominator must contain one 9 and one 0 (yielding 90). This immediately points directly to option (a) without needing to evaluate the rest of the options. Common Pitfall A common error is writing the fraction as (12/99) or (11/99) by ignoring the fact that only the digit 2 repeats while the digit 1 does not. Always differentiate between repeating and fixed decimal positions. Final Answer Therefore, the correct answer is 4(11/90).

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