Question 1

$$ \frac{4+4 	imes 18 - 6 - 8}{123 	imes 6 - 146 	imes 5} = x $$

Accepted Answer

Concept & Logic: This problem requires careful execution of arithmetic operations according to the BODMAS rule, followed by decimal conversion. You must accurately multiply large numbers in the denominator and simplify the entire fraction before finding the exact decimal value. Step-by-Step Solution: • Step 1: Simplify the numerator. Expression: 4 + 4 × 18 - 6 - 8 First, perform the multiplication: 4 × 18 = 72. The expression becomes: 4 + 72 - 6 - 8. Now, process addition and subtraction from left to right: 76 - 6 - 8 = 70 - 8 = 62. Numerator = 62. • Step 2: Simplify the denominator. Expression: 123 × 6 - 146 × 5 Perform the multiplications: 123 × 6 = 738 146 × 5 = 730 Subtract the results: 738 - 730 = 8. Denominator = 8. • Step 3: Form the fraction and solve for the decimal. x = (62/8) Reduce the fraction by dividing by 2: x = (31/4) Convert to decimal: (31/4) = 7 (3/4) = 7.75. Exam Strategy & Shortcut: For 146 × 5, use the shortcut of multiplying by 10 and dividing by 2: 1460 \div 2 = 730. This is much faster than standard vertical multiplication. When you arrive at (62/8), look at the options. You know 8 × 7 = 56 and 8 × 8 = 64. Therefore, the answer must be between 7 and 8. The only option that fits this estimation is 7.75, eliminating the need to do the final long division manually. Common Pitfall: A standard arithmetic error occurs in the denominator calculation 123 × 6. If rushed, students might calculate it as 728 instead of 738, leading to a denominator of -2 and throwing off the entire result. Double-check single-digit multiplications carefully. Final Answer: Therefore, the correct answer is 7.75.

Question 2

Given that $(1^2 + 2^2 + 3^2 + ..... + 10^2) = 385$, the value of $(2^2 + 4^2 + 6^2 + ..... + 20^2)$ is equal to

Accepted Answer

Concept & Logic: This problem tests your ability to recognize algebraic patterns and factor out common multipliers from a series. Instead of calculating the squares manually, look for the relationship between the bases in the two series. Notice that every term in the second series is exactly double the base of the corresponding term in the first series. Step-by-Step Solution: • Step 1: Write out the target series. Target = (2^2 + 4^2 + 6^2 + ..... + 20^2) • Step 2: Express each base as a multiple of the terms from the given series. 2^2 = (2 × 1)^2 = 2^2 × 1^2 4^2 = (2 × 2)^2 = 2^2 × 2^2 6^2 = (2 × 3)^2 = 2^2 × 3^2 ... 20^2 = (2 × 10)^2 = 2^2 × 10^2 • Step 3: Substitute these back into the target series and factor out the common term. Target = (2^2 × 1^2) + (2^2 × 2^2) + (2^2 × 3^2) + ..... + (2^2 × 10^2) Target = 2^2(1^2 + 2^2 + 3^2 + ..... + 10^2) • Step 4: Substitute the given value for the sum of squares. We are given that (1^2 + 2^2 + 3^2 + ..... + 10^2) = 385. Target = 4 × 385 • Step 5: Calculate the final product. 4 × 385 = 1540 Exam Strategy & Shortcut: Whenever you see a series of even squares being compared to a series of consecutive integer squares, remember the multiplier is always 2^2 = 4. You do not need to write out the steps. Just mentally spot that 2^2 factors out, leaving the original series. Immediately multiply the given sum by 4: 385 × 4 = 1540. Common Pitfall: The most common trap is factoring out just 2 instead of 2^2. Students often incorrectly assume that if the bases are doubled, the sum is doubled, leading them to choose 770. Because the terms are squared, the factor factored out must also be squared! Final Answer: Therefore, the correct answer is 1540.

Question 3

$$4\frac{3}{7} - 1\frac{3}{14} = x + 2\frac{3}{28}$$

Accepted Answer

Concept & Strategy: This is a linear equation containing mixed fractions. The strategy is to isolate the variable x on one side of the equation, then separate the resulting expression into whole numbers and fractional parts to simplify efficiently. Notice that the denominators (7, 14, 28) are multiples of each other, making the common denominator obvious. Step-by-Step Solution: • Step 1: Isolate the variable x. Move 2(3/28) to the left side by subtracting it. x = 4(3/7) - 1(3/14) - 2(3/28) • Step 2: Split the expression into whole numbers and fractions. Whole numbers: (4 - 1 - 2) Fractions: ((3/7) - (3/14) - (3/28)) • Step 3: Evaluate the whole numbers. 4 - 1 - 2 = 1 • Step 4: Evaluate the fractions using the LCM of 28. Convert (3/7) to (12/28) Convert (3/14) to (6/28) The fraction expression becomes: (12/28) - (6/28) - (3/28) (12 - 6 - 3/28) = (3/28) • Step 5: Combine the results. Final Answer = 1 + (3/28) = 1(3/28) Exam Strategy & Shortcut: Since the denominators scale perfectly (7 × 2 = 14, 14 × 2 = 28), you can solve the fractions almost entirely in your head. Just think in terms of 28ths: 12 - 6 - 3 = 3. The whole numbers are trivial: 4 - 1 - 2 = 1. The answer assembles itself instantly. Common Pitfall: When shifting 2(3/28) across the equals sign, students sometimes forget to invert the sign of BOTH the whole number and the fraction. Subtracting a mixed number means subtracting the whole part AND subtracting the fractional part. Final Answer: Therefore, the correct answer is 1(3/28).

Question 4

$$6\frac{9}{13} - 4\frac{4}{11} + 2\frac{2}{5} = x$$

Accepted Answer

Concept & Strategy: This problem tests your stamina with fractions involving prime denominators. Because 13, 11, and 5 are all co-prime (they share no common factors), their Least Common Multiple (LCM) is simply their product. As always, separate the whole numbers from the fractions to prevent the numerators from ballooning to unmanageable sizes. Step-by-Step Solution: • Step 1: Separate wholes and fractions. Whole Numbers: 6 - 4 + 2 Fractions: (9/13) - (4/11) + (2/5) • Step 2: Calculate whole numbers. 6 - 4 + 2 = 4 • Step 3: Find the LCM for the fractions. LCM = 13 × 11 × 5 = 143 × 5 = 715. *(Note: All options have a denominator of 715, confirming this step).* • Step 4: Convert each fraction to the common denominator. For (9/13): Multiply by (11 × 5/11 × 5) = (55/55). \rightarrow (9 × 55/715) = (495/715) For (4/11): Multiply by (13 × 5/13 × 5) = (65/65). \rightarrow (4 × 65/715) = (260/715) For (2/5): Multiply by (13 × 11/13 × 11) = (143/143). \rightarrow (2 × 143/715) = (286/715) • Step 5: Combine the fraction numerators. (495 - 260 + 286/715) = (235 + 286/715) = (521/715) • Step 6: Combine wholes and fractions. Result = 4 + (521/715) = 4(521/715) Exam Strategy & Shortcut: Look at the whole numbers first: 6 - 4 + 2 = 4. By simply doing this 2-second calculation, you can immediately eliminate Options C and D (which start with 9). Now you are choosing between 4(596/715) and 4(521/715). You only need to verify the unit digit of the fraction numerator. Numerator unit digit logic: (9 × 5) - (4 × 5) + (2 × 3) \rightarrow 5 - 0 + 6 = 11, which ends in 1. Option B ends in 1 (521). Option A ends in 6 (596). You can confidently choose Option B without doing the full heavy multiplication. Common Pitfall: Calculating the numerators (9 × 55, 4 × 65, etc.) often leads to simple multiplication errors under time pressure. Attempting to solve this problem without using unit-digit elimination on the options takes substantially more time and carries a high risk of arithmetic mistakes. Final Answer: Therefore, the correct answer is 4(521/715).

Question 5

$\frac{1}{(2 \frac{1}{3})} + \frac{1}{(1 \frac{3}{4})}$ is equal to

Accepted Answer

Concept & Strategy: The key insight is converting mixed fractions to improper fractions before taking their reciprocals. Remember that the reciprocal of a fraction is determined by flipping its numerator and denominator. Reciprocal of (a/b) = (b/a) Step-by-Step Solution: • Convert the mixed fractions in the denominators into improper fractions: 2 (1/3) = (2 × 3 + 1/3) = (7/3) 1 (3/4) = (1 × 4 + 3/4) = (7/4) • Substitute these improper fractions back into the original expression: (1/\frac{7){3}} + (1/\frac{7){4}} • Take the reciprocal of each fraction to bring them to the numerator: (3/7) + (4/7) • Since the denominators are identically 7, simply add the numerators together: (3 + 4/7) = (7/7) = 1 • The calculated answer is 1, which does not match options (a), (b), or (c). Exam Strategy & Shortcut: When dealing with expressions structured as "1 over a mixed fraction," immediately convert the mixed fraction to an improper fraction and flip it. Mental math: 2 (1/3) becomes (7/3), flipped is (3/7). 1 (3/4) becomes (7/4), flipped is (4/7). Adding them gives (7/7) = 1. The entire problem can be solved mentally without writing down a single equation. Common Pitfall: Students sometimes incorrectly evaluate the reciprocal of a mixed fraction by only flipping the fractional part (for example, assuming (1/2 \frac{1){3}} is 2 (3/1)). Always convert the entire number to a pure improper fraction before applying the reciprocal. Final Answer: Therefore, the correct answer is None of these.

Question 6

$5 \frac{5}{6} - 3 \frac{8}{9} - x = 1$

Accepted Answer

Concept & Logic: When adding or subtracting mixed fractions, it is mathematically sound and computationally faster to separate the whole integer parts from the fractional parts rather than converting everything into large improper fractions. Step-by-Step Solution: • The given mathematical equation is: 5 (5/6) - 3 (8/9) - x = 1 • Rearrange the equation to isolate the unknown variable x on one side: x = 5 (5/6) - 3 (8/9) - 1 • Group the integer components and the fractional components independently: x = (5 - 3 - 1) + ((5/6) - (8/9)) • Calculate the result of the integer part: 5 - 3 - 1 = 1 • Calculate the fractional part by finding the LCM of 6 and 9, which is 18: (5/6) = (5 × 3/18) = (15/18) (8/9) = (8 × 2/18) = (16/18) • Subtract the fractions: (15/18) - (16/18) = -(1/18) • Combine the processed integer and fractional parts back together: x = 1 - (1/18) = (17/18) Exam Strategy & Shortcut: Split the integers and fractions immediately in your head. 5 - 3 - 1 = 1. The fraction part is (5/6) - (8/9). Multiply cross-wise mentally for rapid subtraction: (45 - 48) / 54 = -3/54 = -1/18. Deducting this from the integer yields 1 - 1/18 = 17/18. Skipping the improper fraction conversion saves massive amounts of time and eliminates arithmetic errors on large numbers. Common Pitfall: Converting 5 (5/6) to (35/6) and 3 (8/9) to (35/9) creates unnecessarily large numerators. This significantly increases the likelihood of calculation errors when finding a common denominator and performing subtraction under time pressure. Final Answer: Therefore, the correct answer is (17/18).

Question 7

$\frac{3}{5}$ of $\frac{4}{7}$ of $\frac{5}{9}$ of $\frac{21}{24}$ of $504 = x$

Accepted Answer

Concept & Strategy: When a problem strings together multiple fractions using the word "of", treat it as one continuous multiplication sequence. The most efficient approach is to simplify everything as a single combined fraction before attempting to multiply by the final whole number. Step-by-Step Solution: • Given expression: (3/5) of (4/7) of (5/9) of (21/24) of 504 • Convert all the "of" operators directly to multiplication: (3/5) × (4/7) × (5/9) × (21/24) × 504 • Isolate the continuous fraction part to cross-cancel and simplify: (3 × 4 × 5 × 21/5 × 7 × 9 × 24) • Begin cancelling terms between the collective numerator and denominator: Cancel the 5 from both the top and bottom. Simplify (4/24) to (1/6). Simplify (21/7) to 3. • The remaining consolidated fraction expression is: (3 × 3/9 × 6) • Further simplify this: (9/9 × 6) = (1/6) • Finally, multiply this resulting fraction by the whole number 504: (1/6) × 504 = (504/6) = 84 • The correctly calculated answer is 84, which is not listed in options (a), (b), (c), or (d). Exam Strategy & Shortcut: Write out the entire multiplication string as a single tall fraction block. Visually cross out pairs immediately: strike through the 5s. Strike through 3 and 4 on top against the 24 on the bottom (since 3 × 4 = 12, and 12 / 24 instantly reduces to 1/2). Then simplify 21 / 7 = 3. You are left with (3 × 1) / (9 × 2) = 3 / 18 = 1/6. From there, evaluating 1/6 of 504 yields 84 in seconds. Common Pitfall: Solving the sequence sequentially from left to right creates messy, unsimplified intermediate fractions (e.g., 12/35). This heavily wastes time and drastically increases the chance of minor arithmetic errors. Always simplify globally across the entire chain instead of locally. Final Answer: Therefore, the correct answer is None of these.

Question 8

$6 \frac{5}{6} 	imes 5 \frac{1}{3} + 17 \frac{2}{3} 	imes 4 \frac{1}{2} = x$

Accepted Answer

Concept & Strategy: When evaluating complex expressions with mixed fractions and multiple operations, strictly follow the BODMAS rule (Bracket, Order, Division, Multiplication, Addition, Subtraction). Before executing operations, convert all mixed fractions into improper fractions for easier arithmetic. Mixed to Improper: a (b/c) = ((a × c) + b/c) Step-by-Step Solution: • First, convert all the mixed numbers in the equation to improper fractions: 6 (5/6) = (41/6) 5 (1/3) = (16/3) 17 (2/3) = (53/3) 4 (1/2) = (9/2) • Substitute these into the original expression: ((41/6) × (16/3)) + ((53/3) × (9/2)) • Calculate the first multiplication block: (41 × 16/6 × 3) = (656/18) = (328/9) • Calculate the second multiplication block: (53 × 9/3 × 2) = (477/6) = (159/2) • Add the two resulting fractions by finding the LCM of the denominators (9 and 2), which is 18: (328 × 2/18) + (159 × 9/18) (656/18) + (1431/18) = (2087/18) • Convert the final improper fraction back to a mixed number: 2087 \div 18 = 115 with a remainder of 17. The exact result is 115 (17/18). • Since 115 (17/18) does not match any of the given options, the answer is "None of these". Exam Strategy & Shortcut: Use rough approximation to verify your path. Evaluate the whole numbers: 7 × 5 = 35, and 18 × 4.5 = 81. Adding them yields 116. This tells you the exact answer is very close to 116. Because the exact fraction is 115 (17/18), you can confidently select "None of these" knowing it slightly misses option (b) which is 116 (2/3). Common Pitfall: Students often incorrectly multiply the whole numbers together and the fractions together separately (e.g., 6 × 5 and (5/6) × (1/3)). This violates the distributive property. Always convert mixed numbers to improper fractions before multiplying. Final Answer: Therefore, the correct answer is None of these.

Question 9

$1 \frac{1}{4} + 1 \frac{5}{9} 	imes 1 \frac{5}{8} \div 6 \frac{1}{2} = x$

Accepted Answer

Concept & Logic: This problem tests strict adherence to the BODMAS order of operations alongside fraction arithmetic. You must resolve Division first, then Multiplication, and finally Addition. Step-by-Step Solution: • The original expression is: 1 (1/4) + 1 (5/9) × 1 (5/8) \div 6 (1/2) • First, convert all mixed numbers to improper fractions: 1 (1/4) = (5/4) 1 (5/9) = (14/9) 1 (5/8) = (13/8) 6 (1/2) = (13/2) • Rewrite the full expression with improper fractions: (5/4) + (14/9) × (13/8) \div (13/2) • Apply BODMAS by solving the division part first ((13/8) \div (13/2)): (13/8) × (2/13) = (2/8) = (1/4) • Now, rewrite the expression with the division solved and proceed to multiplication: (5/4) + ((14/9) × (1/4)) (14/9) × (1/4) = (14/36) = (7/18) • Finally, perform the addition step: (5/4) + (7/18) • Find the Least Common Multiple (LCM) of 4 and 18, which is 36: (5 × 9/36) + (7 × 2/36) = (45/36) + (14/36) = (59/36) • The exact fractional result is (59/36) (or 1 (23/36)). Since all options are whole numbers and do not match, the answer is "None of these". Exam Strategy & Shortcut: A quick glance at the operations and values can save time. You have a small fraction (\approx 1.25) being added to the product of two other small fractions divided by a larger one. The result will unequivocally remain a small mixed decimal/fraction (under 2). Since options (a), (b), (c), and (d) are all large whole numbers (17 or higher), you can instantly mark "None of these" without doing the rigorous fraction arithmetic. Common Pitfall: Ignoring BODMAS and solving left-to-right is the most common path to failure here. Adding the first two terms before multiplying or dividing will yield a completely incorrect and often messy numerical result. Final Answer: Therefore, the correct answer is None of these.

Question 10

$\left( 16 \frac{2}{5} - 12 \frac{1}{15} \right) \div 3 \frac{4}{81} = x$

Accepted Answer

Concept & Formula: This problem requires the application of the BODMAS rule and the conversion of mixed fractions into improper fractions before performing arithmetic operations. a (b/c) = ((a × c) + b/c) Step-by-Step Solution: • Given: ( 16 (2/5) - 12 (1/15) ) \div 3 (4/81) • Convert mixed fractions: • 16 (2/5) = 16 + (6/15) (creating a common denominator) • 12 (1/15) = 12 + (1/15) • Solve the bracket: • ( 16 - 12 ) + ( (6/15) - (1/15) ) • 4 + (5/15) = 4 + (1/3) = (13/3) • Convert the divisor: • 3 (4/81) = ((3 × 81) + 4/81) = (243 + 4/81) = (247/81) • Perform division: • (13/3) \div (247/81) = (13/3) × (81/247) • ( (13/247) ) × ( (81/3) ) • (1/19) × 27 = (27/19) = 1 (8/19) Exam Strategy & Shortcut: Separate the integer part and the fractional part for addition and subtraction. Instead of calculating (16 × 5)+2, calculate (16-12) and ((2/5) - (1/15)) separately. This saves significant calculation time and reduces large number multiplication errors. Common Pitfall: Students often convert 16 (2/5) into (82/5) and 12 (1/15) into (181/15), leading to 3 × 82 = 246 and 246 - 181 = 65, which takes much longer and increases the chance of arithmetic mistakes under pressure. Always split mixed fractions when subtracting. Final Answer: Therefore, the correct answer is None of these.

Question 11

$\frac{3}{2} 	imes \frac{11}{5} \div \left( \frac{25}{44} 	imes \frac{11}{5} ight) \div \frac{33}{15} = x$

Accepted Answer

Concept & Strategy: This simplification problem strictly follows the BODMAS rule. Operations inside parentheses (brackets) must be executed first, followed by continuous division from left to right by multiplying with the reciprocal. Step-by-Step Solution: • Given: (3/2) × (11/5) \div ( (25/44) × (11/5) ) \div (33/15) • Solve the expression inside the bracket: • (25/44) × (11/5) = (5 × 5/4 × 11) × (11/5) = (5/4) • Substitute back and convert division to multiplication: • (3/2) × (11/5) \div (5/4) \div (33/15) • Replace division signs with multiplication by inverting the fractions: • (3/2) × (11/5) × (4/5) × (15/33) • Simplify the terms: • Notice that (3 × 11/33) = (33/33) = 1, so they cancel out completely. • Remaining terms: (1/2) × (1/5) × (4/5) × 15 • (4 × 15/2 × 25) = (60/50) • (60/50) = (6/5) • Convert to mixed fraction: • (6/5) = 1 (1/5) Exam Strategy & Shortcut: Convert all division signs to multiplication immediately after solving brackets. Write it as one long chain of multiplication: (3/2) × (11/5) × (4/5) × (15/33). Cross-cancel aggressively (like 3 × 11 and 33) before doing any actual large multiplication. Common Pitfall: A frequent error in chained division (A \div B \div C) is treating it as A \div (B \div C). Division is left-associative. Always process it left-to-right, or better yet, convert all divisions to multiplication by the reciprocal to avoid ambiguity entirely. Final Answer: Therefore, the correct answer is None of these.

Question 12

$1 + 2 \div \left\{ 1 + 2 \div \left( 1 + \frac{1}{3} \right) \right\}$ is equal to

Accepted Answer

Concept & Strategy: This is a classic simplification problem requiring the strict application of the BODMAS/PEMDAS rule. You must resolve brackets from the innermost parentheses `()` outward to the curly braces `{}` before performing the final arithmetic outside. Step-by-Step Solution: • Given: 1 + 2 \div { 1 + 2 \div ( 1 + (1/3) ) } • Step 1: Solve the innermost parentheses • 1 + (1/3) = (3/3) + (1/3) = (4/3) • The expression becomes: 1 + 2 \div { 1 + 2 \div (4/3) } • Step 2: Solve the division inside the curly braces • Convert division to multiplication by inverting the fraction: 2 \div (4/3) = 2 × (3/4) • 2 × (3/4) = (6/4) = (3/2) • The expression becomes: 1 + 2 \div { 1 + (3/2) } • Step 3: Solve the addition inside the curly braces • 1 + (3/2) = (2/2) + (3/2) = (5/2) • The expression becomes: 1 + 2 \div (5/2) • Step 4: Final division and addition • Division comes before addition: 2 \div (5/2) = 2 × (2/5) = (4/5) • Finally, add to 1: 1 + (4/5) = 1 (4/5) Exam Strategy & Shortcut: Write the expression out and solve it "in place" by crossing out resolved sections. Once you see (1 + (1/3)) is (4/3), immediately flip it and multiply by 2 to get (3/2) mentally. Adding 1 gives (5/2). Flip it again and multiply by 2 to get (4/5). Adding 1 gives 1 (4/5). This requires almost zero scratchpad writing. Common Pitfall: A very common mistake is ignoring the order of operations and adding the 1 + 2 at the very beginning to make it 3 \div \{ \dots \}. Always remember that addition is the absolute last step unless it is contained within a bracket. Final Answer: Therefore, the correct answer is 1 (4/5).

Question 13

Simplify: $10 \frac{1}{8} 	ext{ of } \frac{12}{15} \div \frac{35}{36} 	ext{ of } \frac{20}{49}$.

Accepted Answer

Concept & Rule: This problem tests your exact understanding of the BODMAS hierarchy. The word 'of' translates to multiplication, but critically, operations connected by 'of' must be calculated before division. A of B \div C of D = (A × B) \div (C × D) Step-by-Step Solution: • Given: 10 (1/8) of (12/15) \div (35/36) of (20/49) • Step 1: Convert mixed numbers and evaluate the 'of' terms on both sides • Left side: 10 (1/8) = (81/8) • Left 'of' evaluation: (81/8) × (12/15) • Simplify by cross-canceling (12 and 15 by 3 to get (4/5)): (81/8) × (4/5) = (81/2) × (1/5) = (81/10) • Right side 'of' evaluation: (35/36) × (20/49) • Simplify by cross-canceling (divide 35 and 49 by 7; divide 20 and 36 by 4): • (5/9) × (5/7) = (25/63) • Step 2: Perform the central division • Now substitute the simplified blocks back into the equation: (81/10) \div (25/63) • Convert division to multiplication by flipping the second fraction: (81/10) × (63/25) • Multiply numerators and denominators: (81 × 63/10 × 25) = (5103/250) • Step 3: Convert the improper fraction back to a mixed number • 5103 \div 250: We know 250 × 20 = 5000. • The remainder is 103. • Result: 20 (103/250) Exam Strategy & Shortcut: Treat everything before the \div sign as one massive bracket, and everything after as another massive bracket. Simplify each block independently through aggressive cross-cancellation before attempting to divide. Never multiply large numerators (like 35 × 20) without checking for common factors first. Common Pitfall: The most frequent error is treating 'of' identically to a standard multiplication sign (×) and processing strictly left-to-right: (10 (1/8) × (12/15)) \div ((35/36)) × ((20/49)). This completely changes the mathematical meaning and will result in a totally incorrect fraction. Final Answer: Therefore, the correct answer is 20 (103/250).

Question 14

$\frac{-\frac{1}{2} - \frac{2}{3} + \frac{4}{5} - \frac{1}{3} + \frac{1}{5} + \frac{3}{4}}{\frac{1}{2} + \frac{2}{3} - \frac{4}{3} + \frac{1}{3} - \frac{1}{5} - \frac{4}{5}}$ is simplified to

Accepted Answer

Concept & Strategy: When faced with a long chain of fractional additions and subtractions, do not immediately try to find a massive Least Common Multiple (LCM) for all terms. Instead, group fractions that share the same denominator to simplify the expression into small integers before doing any final cross-multiplication. Step-by-Step Solution: • Given Expression: • Numerator: -(1/2) - (2/3) + (4/5) - (1/3) + (1/5) + (3/4) • Denominator: (1/2) + (2/3) - (4/3) + (1/3) - (1/5) - (4/5) • Step 1: Group and simplify the Numerator terms • Group by denominator 2 & 4: (-(1/2) + (3/4)) = (-(2/4) + (3/4)) = (1/4) • Group by denominator 3: (-(2/3) - (1/3)) = -(3/3) = -1 • Group by denominator 5: ((4/5) + (1/5)) = (5/5) = 1 • Total Numerator = (1/4) - 1 + 1 = (1/4) • Step 2: Group and simplify the Denominator terms • Keep denominator 2: (1/2) • Group by denominator 3: ((2/3) - (4/3) + (1/3)) = (2 - 4 + 1/3) = -(1/3) • Group by denominator 5: (-(1/5) - (4/5)) = -(5/5) = -1 • Total Denominator = (1/2) - (1/3) - 1 = (3 - 2/6) - 1 = (1/6) - (6/6) = -(5/6) • Step 3: Divide the simplified Numerator by the Denominator • (\frac{1/4)}{-(5/6)} = (1/4) \div -(5/6) • (1/4) × -(6/5) = -(6/20) • Simplify by dividing by 2: -(3/10) Exam Strategy & Shortcut: Scan the equation quickly. Notice how +(4/5) and +(1/5) cleanly make a 1, and -(2/3) and -(1/3) make a -1. These cancel each other out instantly, leaving only -(1/2) + (3/4) in the numerator. Visual grouping like this turns a 2-minute calculation into a 20-second visual puzzle. Common Pitfall: The worst mistake you can make here is finding the LCM for 2, 3, 4, 5 (which is 60) and multiplying every single fraction's numerator. E.g., (-(30/60) - (40/60) + (48/60) \dots). This creates unnecessarily large numbers and almost guarantees a basic arithmetic or negative sign error under time pressure. Always look for natural groupings first. Final Answer: Therefore, the correct answer is -(3/10).

Question 15

When $\left( \frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \right)$ is divided by $\left( \frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18} \right)$, the result is

Accepted Answer

Concept & Strategy: To divide two complex bracketed expressions, solve each bracket independently first. When finding the Least Common Multiple (LCM) for multiple fractions, check if any fractions share a denominator or have easily combinable values to save calculation time. Step-by-Step Solution: • Evaluate the first bracket (Numerator): • Expression: (1/2) - (1/4) + (1/5) - (1/6) • Find the LCM of 2, 4, 5, and 6, which is 60. • Convert fractions: (30/60) - (15/60) + (12/60) - (10/60) • Combine: (30 - 15 + 12 - 10/60) = (17/60) • Evaluate the second bracket (Denominator): • Expression: (2/5) - (5/9) + (3/5) - (7/18) • Group fractions with the same or related denominators to simplify. • Group the 5s: (2/5) + (3/5) = (5/5) = 1 • Group the 9 and 18: -(5/9) - (7/18) • Find common denominator (18): -(10/18) - (7/18) = -(17/18) • Combine the groups: 1 - (17/18) = (1/18) • Perform the final division: • (17/60) \div (1/18) • Convert division to multiplication by inverting the second fraction: (17/60) × 18 • Simplify by dividing 18 and 60 by 6: (17 × 3/10) = (51/10) • Convert to mixed fraction: 5 (1/10) Exam Strategy & Shortcut: Instantly spot that (2/5) + (3/5) = 1 in the divisor. This turns the messy second bracket into a trivial 1 - (17/18) = (1/18). Doing this visually removes half the workload of the problem, leaving only the first bracket to calculate properly. Common Pitfall: Attempting to find a single massive LCM (like 90) for the entire second bracket without grouping first. Calculating (36/90) - (50/90) + (54/90) - (35/90) is much more prone to arithmetic errors under time pressure than simply pairing off the obvious matches. Final Answer: Therefore, the correct answer is 5 (1/10).

Question 16

Find the value of $x$ in the following  $$1\frac{2}{3} \div \frac{2}{7} 	imes \frac{x}{7} = 1\frac{1}{4} 	imes \frac{2}{3} \div \frac{1}{6}$$

Accepted Answer

Concept & Formula The core concept involves following the correct order of operations (BODMAS/PEMDAS) and handling operations on mixed fractions. The key rule for dividing fractions is to multiply by their reciprocal: (a/b) \div (c/d) = (a/b) × (d/c). Step-by-Step Solution • First, convert the mixed fractions into improper fractions: 1(2/3) = (5/3) 1(1/4) = (5/4) • Substitute these into the original equation: (5/3) \div (2/7) × (x/7) = (5/4) × (2/3) \div (1/6) • Apply the division-to-multiplication rule (invert the fraction immediately following the division sign) on both sides: LHS: (5/3) × (7/2) × (x/7) = (5 × 7 × x/3 × 2 × 7) = (5x/6) RHS: (5/4) × (2/3) × (6/1) = (5 × 2 × 6/4 × 3 × 1) = (60/12) = 5 • Equate LHS and RHS: (5x/6) = 5 • Solve for x: 5x = 30 \implies x = 6 Exam Strategy & Shortcut Convert mixed numbers to improper fractions immediately. Cancel common terms from the numerators and denominators *before* multiplying them out. For example, on the LHS, the 7 in the numerator and denominator cancel out instantly, saving calculation time. Common Pitfall A frequent error is misinterpreting the order of operations by performing multiplication before division. Always process division and multiplication strictly from left to right as they hold the same mathematical precedence. Final Answer Therefore, the correct answer is 6.

Question 17

The difference of $1\frac{3}{16}$ and its reciprocal is equal to

Accepted Answer

Concept & Formula The reciprocal of a fraction (a/b) is (b/a). To find the exact difference between a mixed fraction and its reciprocal, you must first convert the mixed fraction into an improper fraction before finding a common denominator to subtract them. Step-by-Step Solution • First, convert the mixed fraction into an improper fraction: 1(3/16) = (19/16) • Find the reciprocal of this improper fraction: Reciprocal = (16/19) • Calculate the difference by subtracting the smaller value from the larger value: Difference = (19/16) - (16/19) • Find the common denominator (16 × 19 = 304) and cross-multiply to subtract the numerators: (19^2 - 16^2/304) • Simplify using the difference of squares algebraic identity (a^2 - b^2 = (a-b)(a+b)): 19^2 - 16^2 = (19 - 16)(19 + 16) = 3 × 35 = 105 • The exact difference is (105/304). • Since this calculated fraction is not present among any of the given specific options, the correct choice is "None of these". Exam Strategy & Shortcut Use the algebraic identity a^2 - b^2 = (a-b)(a+b) to evaluate the numerator quickly. Calculating (19-16) × (19+16) mentally is vastly faster than manually squaring 19 and 16 and then subtracting them (361 - 256). Common Pitfall A prevalent mistake is finding the reciprocal incorrectly by only inverting the fractional part of the mixed number (e.g., treating the reciprocal of 1(3/16) as 1(16/3)). Always convert fully to an improper fraction before flipping the numerator and denominator. Final Answer Therefore, the correct answer is None of these.

Question 18

If $	ext{I} = \frac{3}{4} \div \frac{5}{6}$, $	ext{II} = 3 \div [(4 \div 5) \div 6]$, $	ext{III} = [3 \div (4 \div 5)] \div 6$, $	ext{IV} = 3 \div 4 \div (5 \div 6)$, then

Accepted Answer

Concept & Logic This problem evaluates how the placement of brackets alters the outcome of chained division operations. The core principle remains: division must follow BODMAS, executing operations inside brackets first, and transforming division by a fraction into multiplication by its reciprocal. Step-by-Step Solution • Evaluate Expression I: I = (3/4) \div (5/6) = (3/4) × (6/5) = (18/20) = (9/10) • Evaluate Expression II: II = 3 \div [((4/5)) \div 6] II = 3 \div [(4/5) × (1/6)] = 3 \div [(4/30)] II = 3 × (30/4) = (90/4) = (45/2) • Evaluate Expression III: III = [3 \div ((4/5))] \div 6 III = [3 × (5/4)] \div 6 = (15/4) \div 6 III = (15/4) × (1/6) = (15/24) = (5/8) • Evaluate Expression IV: IV = 3 \div 4 \div ((5/6)) Evaluate left to right first: (3 \div 4) = (3/4) IV = (3/4) \div (5/6) = (3/4) × (6/5) = (18/20) = (9/10) • By comparing the simplified results, we see that I = (9/10) and IV = (9/10). Exam Strategy & Shortcut Notice that Expression IV is 3 \div 4 \div ((5/6)). Due to the left-to-right rule, 3 \div 4 groups naturally into (3/4). The expression essentially becomes (3/4) \div (5/6), which is identical character-for-character to Expression I. Recognizing this structural identicality saves you from having to calculate II and III entirely. Common Pitfall Assuming division is associative is a critical error. Evaluating Expression IV as 3 \div (4 \div (5/6)) instead of (3 \div 4) \div (5/6) will result in a completely incorrect answer. Final Answer Therefore, the correct answer is I and IV are equal.

Question 19

$$2\frac{3}{4} \div 2\frac{2}{3} \div 1\frac{1}{12} = x$$

Accepted Answer

Concept & Formula When handling a chain of continuous division operations with mixed numbers, the standard procedure involves two steps: first, convert every mixed number into an improper fraction. Second, process the chain strictly from left to right, which is mathematically achieved by changing every division sign into a multiplication sign while simultaneously taking the reciprocal of the fraction immediately following it. Step-by-Step Solution • Step 1: Convert all mixed numbers to improper fractions. 2(3/4) = (4 × 2 + 3/4) = (11/4) 2(2/3) = (3 × 2 + 2/3) = (8/3) 1(1/12) = (12 × 1 + 1/12) = (13/12) • Step 2: Rewrite the original equation. (11/4) \div (8/3) \div (13/12) = x • Step 3: Convert divisions to multiplications. Change each \div to × and flip the following fraction: x = (11/4) × (3/8) × (12/13) • Step 4: Simplify and Calculate. Multiply the numerators together and the denominators together: x = (11 × 3 × 12/4 × 8 × 13) Cancel the common factor of 4 between the numerator (12) and denominator (4): x = (11 × 3 × 3/1 × 8 × 13) x = (99/104) • Step 5: Verify against options. The calculated value is (99/104). None of the specific options (a, b, c) match this value or can be simplified to this value. Exam Strategy & Shortcut Write the expression as a single large fraction immediately: (11 × 3 × 12/4 × 8 × 13). Perform cross-cancellations before executing any multiplication. This prevents you from dealing with unnecessarily large numbers like (396/416), which are highly prone to calculation errors during simplification. Common Pitfall The most dangerous pitfall is applying division associatively. If a student calculates the right side first (2(2/3) \div 1(1/12)) and then divides 2(3/4) by that result, they will arrive at an entirely wrong answer. Continuous division must always be executed from left to right. Final Answer Therefore, the correct answer is None of these.

Question 20

$$\frac{4335}{4(x)24} \div 1\frac{7}{8} = \frac{289}{528}$$

Accepted Answer

Concept & Strategy This question involves identifying a missing digit in a 4-digit number (represented as 4(x)24) within an equation. The optimal strategy is not cross-multiplication immediately, but rather algebraic simplification. Isolate the term containing the unknown digit and use cross-cancellation to reveal the hidden value. Step-by-Step Solution • Step 1: Simplify the known elements Convert the mixed fraction to an improper fraction: 1(7/8) = (15/8) Rewrite the equation: (4335/4(x)24) \div (15/8) = (289/528) • Step 2: Convert division to multiplication Multiply by the reciprocal of the divisor: (4335/4(x)24) × (8/15) = (289/528) • Step 3: Isolate the unknown term Multiply both sides by (15/8) to isolate the fraction containing the unknown: (4335/4(x)24) = (289/528) × (15/8) (4335/4(x)24) = (289 × 15/4224) • Step 4: Cancel common factors Check if the numerators relate to each other to simplify. Let's divide 4335 by 15: 4335 \div 15 = 289 Therefore, the equation simplifies perfectly: (289 × 15/4(x)24) = (289 × 15/4224) • Step 5: Compare denominators Since the numerators are identical (289 × 15), the denominators must also be identical: 4(x)24 = 4224 By direct comparison, the missing digit x is 2. Exam Strategy & Shortcut In competitive exams, numbers are chosen to cancel out. Notice the 289 on the right side. Before multiplying large numbers, test if the large numerator on the left (4335) is a multiple of 289. 289 × 15 = 4335. This immediate recognition allows you to cancel the numerators almost instantly, turning a complex arithmetic problem into a simple visual comparison. Common Pitfall A major time-trap is attempting to cross-multiply everything immediately: 4335 × 8 × 528 = 289 × 15 × 4(x)24. This results in massive numbers that are highly prone to calculation errors. Always simplify and cancel horizontally before cross-multiplying. Final Answer Therefore, the correct answer is 2.

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