Q: Solve $\left(\frac{18}{4}\right)^2 \times \left(\frac{455}{19}\right) \div \left(\frac{61}{799}\right) = x$

Concept & Logic: This is a classic approximation problem found in banking exams. The numbers are deliberately designed to be slightly off from clean multiples. The strategy is to round the tricky fractions to the nearest whole numbers or simple ratios to execute the calculation quickly. Step-by-Step Solution: • Step 1: Simplify the first term. • (18/4) = 4.5. • (4.5)^2 = 20.25. • Step 2: Approximate the second term. • Notice that 19 \cdot 24 = 456. • Therefore, (455/19) is extremely close to (456/19) = 24. • Step 3: Approximate the third term (the divisor). • Notice that 61 \cdot 13 = 793. So (799/61) is slightly more than 13. • Alternatively, for an easier mental division inversion, (61/799) \approx (60/800) = (3/40). • Let's use the exact inversion: \div ((61/799)) becomes \cdot ((799/61)). • As established, (799/61) \approx 13.1. • Step 4: Multiply the approximated values together. • x \approx 20.25 \cdot 24 \cdot 13.1 • 20.25 \cdot 24 = 486. (Calculated easily as 20 \cdot 24 = 480, and 0.25 \cdot 24 = 6). • 486 \cdot 13.1 \approx 486 \cdot 13 + 486 \cdot 0.1 • 486 \cdot 13 = 6318. • 6318 + 48.6 = 6366.6. • The closest option provided by the exam setter for this approximation is 6350. (If calculated without any rounding, the exact value is roughly 6351.6). Exam Strategy & Shortcut: In approximation questions with options spread 30-50 units apart, immediately replace ugly fractions with their closest clean integers. (455/19) \rightarrow 24. For (799/61), flip it to multiplication: (800/61) \approx 13.1. Then execute 20.25 \cdot 24 \cdot 13.1. Common Pitfall: The biggest mistake is trying to calculate the exact decimal values of (455/19) and (61/799). This will consume minutes of your time and likely lead to an arithmetic error, whereas approximation takes seconds. Final Answer: Therefore, the correct answer is 6350.

Question 1

Solve $48.2 	imes 2.5 	imes 2.2 + x = 270$

Accepted Answer

Concept & Strategy Group decimal multiplications strategically. Instead of multiplying sequentially from left to right, look for pairs that form whole numbers or simpler decimals to minimize arithmetic errors. Step-by-Step Solution • Given equation: 48.2 × 2.5 × 2.2 + x = 270 • First, group the second and third terms to simplify the multiplication: 2.5 × 2.2 • Break 2.2 into 2 × 1.1 for easier mental math: 2.5 × 2 = 5 5 × 1.1 = 5.5 • Now multiply this result by the first term: 48.2 × 5.5 • Use the fraction equivalent for 5.5 (which is (11/2)): 48.2 × (11/2) = 24.1 × 11 24.1 × 11 = 265.1 • Substitute this back into the original equation: 265.1 + x = 270 x = 270 - 265.1 = 4.9 Exam Strategy & Shortcut Always scan for factors of 5 and even numbers. Spotting that 2.5 × 2.2 easily collapses into 5.5 saves a massive amount of manual decimal multiplication. Additionally, converting 5.5 to (11/2) allows you to divide by 2 rather than multiply by a decimal, which is vastly faster and safer. Common Pitfall Multiplying sequentially (48.2 × 2.5) creates large intermediate values (120.5) that are cumbersome to multiply by 2.2 in the next step, drastically increasing the chance of calculation mistakes. Final Answer Therefore, the correct answer is 4.9.

Question 2

Solve $1599 \div 39.99 + \frac{4}{5} 	imes 2449 - 120.05 = x$

Accepted Answer

Concept & Strategy This is an approximation problem, signaled by the complex decimals and widely spaced answer choices. Round decimals to their nearest whole numbers to drastically speed up calculation while maintaining sufficient accuracy. Step-by-Step Solution • Approximate the given values to their nearest integers: 1599 \approx 1600 39.99 \approx 40 2449 \approx 2450 120.05 \approx 120 • Rewrite the equation using BODMAS rules (Division and Multiplication before Addition and Subtraction): 1600 \div 40 + (4/5) × 2450 - 120 = x • Perform the division: 1600 \div 40 = 40 • Perform the multiplication: (4/5) × 2450 = 4 × 490 = 1960 • Substitute the results back into the simplified equation: 40 + 1960 - 120 2040 - 120 = 1880 Exam Strategy & Shortcut Recognize the approximation cues instantly. 1599 \div 39.99 is intentionally designed to look ugly but perfectly rounds to 1600 \div 40 = 40. Treat (4/5) × 2449 as (4/5) × 2450. Mentally calculate 2450 \div 5 = 490, then 490 × 4 = 1960. Add 40 to get 2000, and drop the 120. The entire problem can be mapped out mentally in under 15 seconds. Common Pitfall Attempting to calculate exact values (like finding the precise result of (4/5) × 2449 = 1959.2) wastes immense amounts of time and creates messy decimals that often lead to addition/subtraction errors in the final steps. Never do exact math on an approximation question. Final Answer Therefore, the correct answer is 1880.

Question 3

Solve $1576 \div 45.02 + 23.99 	imes \sqrt{255} = x$

Accepted Answer

Concept & Strategy: The problem involves complex decimal numbers and square roots. The key insight is to recognize this as an Approximation problem. Use the BODMAS rule alongside rounding numbers to their nearest manageable integers. Step-by-Step Solution: • Given Equation: 1576 \div 45.02 + 23.99 × \sqrt{255} = x • Approximation Step: Round the numbers to the nearest integers or perfect squares to simplify calculations: • 1576 can be rounded to 1575 (since it is easily divisible by 45). • 45.02 rounds to 45. • 23.99 rounds to 24. • \sqrt{255} is extremely close to \sqrt{256}, which is exactly 16. • Calculation: Substituting the approximate values back into the equation: x \approx 1575 \div 45 + 24 × 16 Using BODMAS, perform division and multiplication first: 1575 \div 45 = 35 24 × 16 = 384 Add the results together: x \approx 35 + 384 = 419 • Final Match: The calculated value 419 is closest to option 420. Exam Strategy & Shortcut: In competitive exams, decimal values ending in `.99` or `.02` are glaring signals to use approximation. Instead of doing exact division, estimate 1576 \div 45 by thinking 45 × 10 = 450, 45 × 30 = 1350, remaining is roughly 225 which is 5 × 45. Total 35. Calculate 24 × 16 mentally as (20+4)× 16 = 320 + 64 = 384. Sum is 419 \approx 420. Common Pitfall: Attempting to calculate the exact decimal values will consume a massive amount of time. Another mistake is ignoring the BODMAS rule and adding terms before finishing the division or multiplication. Final Answer: Therefore, the correct answer is 420.

Question 4

Solve $2\frac{2}{9} + 4\frac{1}{18} - 1\frac{1}{2} = x$

Accepted Answer

Concept & Formula: When adding or subtracting mixed fractions, the fastest method is to separate the whole numbers from the fractional parts. A(b/c) + D(e/f) = (A + D) + ((b/c) + (e/f)) Step-by-Step Solution: • Given Equation: 2(2/9) + 4(1/18) - 1(1/2) = x • Step 1: Separate Whole Numbers and Fractions Group the integers and the proper fractions separately: x = (2 + 4 - 1) + ((2/9) + (1/18) - (1/2)) • Step 2: Simplify the Whole Numbers 2 + 4 - 1 = 5 • Step 3: Simplify the Fractions Find the Lowest Common Multiple (LCM) of the denominators 9, 18, and 2, which is 18. Convert all fractions to have the denominator 18: (2/9) = (4/18) (1/18) = (1/18) (1/2) = (9/18) Now, evaluate the fraction part: (4/18) + (1/18) - (9/18) = (4 + 1 - 9/18) = (-4/18) = -(2/9) • Step 4: Combine Both Parts x = 5 - (2/9) To subtract the fraction, borrow 1 from 5: x = 4 + 1 - (2/9) = 4 + (9 - 2/9) = 4(7/9) Exam Strategy & Shortcut: Never convert mixed fractions into improper fractions for addition/subtraction (e.g., 20/9 + 73/18 - 3/2). That balloons the numbers and invites calculation errors. Always handle the integers first (2+4-1=5), then quickly balance the fraction denominators mentally. Common Pitfall: Students often convert mixed fractions to improper fractions (like 20/9), which makes finding common denominators and the final division much more tedious and error-prone. Another mistake is improperly handling negative fractional remainders. Final Answer: Therefore, the correct answer is 4(7/9).

Question 5

Solve $\frac{294 \div 14 	imes 5 + 11}{x} = 8^2 \div 5 + 1.7$

Accepted Answer

Concept & Logic: This problem requires strict adherence to the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction). The equation must be simplified by evaluating the numerator on the left and the entire expression on the right before isolating x. Step-by-Step Solution: • Given Equation: (294 \div 14 × 5 + 11/x) = 8^2 \div 5 + 1.7 • Step 1: Simplify the Left-Hand Side (LHS) Numerator According to BODMAS, perform division first, then multiplication, then addition. 294 \div 14 = 21 21 × 5 = 105 105 + 11 = 116 So, the LHS becomes (116/x). • Step 2: Simplify the Right-Hand Side (RHS) Resolve the exponent (order) first, then division, then addition. 8^2 = 64 64 \div 5 = 12.8 12.8 + 1.7 = 14.5 So, the RHS becomes 14.5. • Step 3: Solve for x Equate the simplified sides: (116/x) = 14.5 x = (116/14.5) To remove the decimal, multiply the numerator and denominator by 10: x = (1160/145) Dividing gives: x = 8 Exam Strategy & Shortcut: Simplify expressions in chunks mentally. For LHS: 294/14 = 21; 21 × 5 = 105; 105+11 = 116. For RHS: 64/5 = 12.8; 12.8+1.7 = 14.5. Now you have 116 = 14.5 × x. Instead of long division, check the unit digits or use multiples. 14.5 × 2 = 29. 29 × 4 = 116. Therefore, x = 2 × 4 = 8. Common Pitfall: Failing to follow BODMAS in the numerator is the most common mistake (e.g., multiplying before dividing or adding before multiplying). Also, arithmetic errors frequently happen when dividing by decimals like 14.5. Final Answer: Therefore, the correct answer is 8.

Question 6

Solve $2.5 	imes 4.8 + 7.2 	imes 1.5 - 1.2 	imes 14 = x$

Accepted Answer

Concept & Logic: The problem requires applying the standard order of operations (BODMAS/PEMDAS) to decimal arithmetic. Multiplication must be performed before addition and subtraction. Step-by-Step Solution: • Step 1: Isolate and perform the multiplication operations first. • 2.5 × 4.8 = 12 • 7.2 × 1.5 = 10.8 • 1.2 × 14 = 16.8 • Step 2: Substitute these products back into the original equation. • Equation becomes: 12 + 10.8 - 16.8 • Step 3: Perform addition and subtraction from left to right. • 22.8 - 16.8 = 6 Exam Strategy & Shortcut: To multiply decimals quickly, temporarily ignore the decimal points, perform integer multiplication, and then place the decimal back based on the total decimal places. For example, for 2.5 × 4.8, calculate 25 × 48. An easy way to do 25 × 48 is 100/4 × 48 = 1200. Since there are two decimal places in total (2.5 and 4.8), shift the decimal two spots to the left: 12.00. Common Pitfall: Students often make alignment errors when adding or subtracting the final terms, such as subtracting 16.8 from 10.8 incorrectly or misplacing the decimal point during the initial multiplication steps. Final Answer: Therefore, the correct answer is 6.

Question 7

Which of the following fractions is the largest?  $\frac{3}{2}, \frac{7}{3}, \frac{5}{4}, \frac{7}{2}$

Accepted Answer

Concept & Logic: To compare fractions efficiently, you can convert them into decimal values or compare their numerators and denominators if they share common traits. Since the denominators here are small (2, 3, 4), converting them directly to decimals is the fastest approach. Step-by-Step Solution: • Step 1: Convert each fraction into its decimal equivalent. • (3/2) = 1.5 • (7/3) = 2.33... • (5/4) = 1.25 • (7/2) = 3.5 • Step 2: Compare the resulting decimal values. • The sequence from smallest to largest is: 1.25 < 1.5 < 2.33 < 3.5. • Step 3: Identify the largest decimal and its corresponding fraction. • The largest decimal is 3.5, which corresponds to (7/2). Exam Strategy & Shortcut: You can instantly eliminate (3/2) and (5/4) by observing the numerators and denominators. (7/2) is exactly half of 7, meaning it is larger than 3. The other fractions ((3/2), (7/3), (5/4)) are all clearly smaller than 3. No calculation is needed if you rely on basic estimation. Common Pitfall: Students sometimes resort to finding a common denominator (LCM of 2, 3, 4, 2 is 12) to cross-compare. While technically correct, cross-multiplication or finding LCMs takes much longer than simple decimal conversion or logical elimination for small numbers. Final Answer: Therefore, the correct answer is (7/2).

Question 8

Solve $\left(\frac{18}{4}ight)^2 	imes \left(\frac{455}{19}ight) \div \left(\frac{61}{799}ight) = x$

Accepted Answer

Concept & Logic: This is a classic approximation problem found in banking exams. The numbers are deliberately designed to be slightly off from clean multiples. The strategy is to round the tricky fractions to the nearest whole numbers or simple ratios to execute the calculation quickly. Step-by-Step Solution: • Step 1: Simplify the first term. • (18/4) = 4.5. • (4.5)^2 = 20.25. • Step 2: Approximate the second term. • Notice that 19 \cdot 24 = 456. • Therefore, (455/19) is extremely close to (456/19) = 24. • Step 3: Approximate the third term (the divisor). • Notice that 61 \cdot 13 = 793. So (799/61) is slightly more than 13. • Alternatively, for an easier mental division inversion, (61/799) \approx (60/800) = (3/40). • Let's use the exact inversion: \div ((61/799)) becomes \cdot ((799/61)). • As established, (799/61) \approx 13.1. • Step 4: Multiply the approximated values together. • x \approx 20.25 \cdot 24 \cdot 13.1 • 20.25 \cdot 24 = 486. (Calculated easily as 20 \cdot 24 = 480, and 0.25 \cdot 24 = 6). • 486 \cdot 13.1 \approx 486 \cdot 13 + 486 \cdot 0.1 • 486 \cdot 13 = 6318. • 6318 + 48.6 = 6366.6. • The closest option provided by the exam setter for this approximation is 6350. (If calculated without any rounding, the exact value is roughly 6351.6). Exam Strategy & Shortcut: In approximation questions with options spread 30-50 units apart, immediately replace ugly fractions with their closest clean integers. (455/19) \rightarrow 24. For (799/61), flip it to multiplication: (800/61) \approx 13.1. Then execute 20.25 \cdot 24 \cdot 13.1. Common Pitfall: The biggest mistake is trying to calculate the exact decimal values of (455/19) and (61/799). This will consume minutes of your time and likely lead to an arithmetic error, whereas approximation takes seconds. Final Answer: Therefore, the correct answer is 6350.

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