Question 1

The value of $$ \left( \frac{0.125 + 0.027}{0.5 	imes 0.5 + 0.09 - 0.15} ight) $$ is

Accepted Answer

Concept & Formula: This problem hides a classic algebraic identity behind raw decimal numbers. The core concept is recognizing perfect cubes and using the sum of cubes expansion. The required formula is: a^3 + b^3 = (a + b)(a^2 - ab + b^2) Step-by-Step Solution: Observe the numbers in the numerator. They are perfect cubes: • 0.125 = 0.5 × 0.5 × 0.5 = (0.5)^3 • 0.027 = 0.3 × 0.3 × 0.3 = (0.3)^3 Let a = 0.5 and b = 0.3. The numerator translates to a^3 + b^3. Now examine the denominator to see if it matches the a^2 - ab + b^2 pattern: • 0.5 × 0.5 represents a^2. • 0.09 is exactly (0.3)^2, which represents b^2. • 0.15 is exactly 0.5 × 0.3, which represents ab. The denominator perfectly matches the form a^2 + b^2 - ab. Write the full expression algebraically: (a^3 + b^3/a^2 - ab + b^2) Expand the numerator using the identity: ((a + b)(a^2 - ab + b^2)/a^2 - ab + b^2) The quadratic term cancels out from the top and bottom, leaving only (a + b). Substitute the values back: 0.5 + 0.3 = 0.8 Exam Strategy & Shortcut: When you spot cubes like 0.125 and 0.027 in the numerator over a mixed quadratic form, instantly extract the cube roots: 0.5 and 0.3. If the sign in the numerator is positive, simply add the roots: 0.5 + 0.3 = 0.8. You can solve this entirely in your head in under 5 seconds. Common Pitfall: A frequent mistake is misidentifying the cube root of decimals. Many students hastily assume the cube root of 0.027 is 0.03 instead of 0.3. Always double-check decimal places: 0.3 × 0.3 × 0.3 yields three decimal places (0.027). Final Answer: Therefore, the correct answer is 0.8.

Question 2

$$ \left[ \frac{8 (3.75)^3 + 1}{(7.5)^2 - 6.5} \right] $$ is equal to:

Accepted Answer

Concept & Formula: This is a heavily disguised sum of cubes problem. The challenge is using the properties of exponents to manipulate the numerator and denominator into the standard a^3 + b^3 identity format. The core formulas are: (xy)^n = x^n y^n a^3 + b^3 = (a + b)(a^2 - ab + b^2) Step-by-Step Solution: Let's resolve the numerator first to reveal the true value of our variable a. We know that 8 = 2^3. Rewrite the first term in the numerator: 8(3.75)^3 = 2^3 × (3.75)^3 Using exponent rules, combine the bases: (2 × 3.75)^3 = (7.5)^3 So, the numerator is actually (7.5)^3 + 1^3. This tells us that a = 7.5 and b = 1. Now, let's verify if the denominator matches the a^2 - ab + b^2 pattern required for the shortcut: a^2 - ab + b^2 = (7.5)^2 - (7.5 × 1) + 1^2 = (7.5)^2 - 7.5 + 1 = (7.5)^2 - 6.5 The calculated pattern perfectly matches the given denominator! The entire expression represents: (a^3 + b^3/a^2 - ab + b^2) Which simplifies to (a + b). Calculate the final result: 7.5 + 1 = 8.5 Exam Strategy & Shortcut: Test setters love masking numbers. Whenever you see a coefficient like 8 or 27 next to a cubed term, instantly pull it inside the parentheses as its cube root (2 or 3). Doing 2 × 3.75 = 7.5 immediately reveals the hidden variable a. From there, 7.5 + 1 = 8.5 is a straightforward mental step. Common Pitfall: Attempting to compute 3.75^3 manually or trying to expand 7.5^2. This question is designed specifically to punish brute-force arithmetic. If a calculation looks impossibly tedious, step back—you are missing an algebraic substitution. Final Answer: Therefore, the correct answer is 8.5.

Question 3

$$ \frac{(4.53 - 3.07)^2}{(3.07 - 2.15)(2.15 - 4.53)} + \frac{(3.07 - 2.15)^2}{(2.15 - 4.53)(4.53 - 3.07)} + \frac{(2.15 - 4.53)^2}{(4.53 - 3.07)(3.07 - 2.15)} $$  is simplified to

Accepted Answer

Concept & Formula: This problem leverages a powerful conditional algebraic identity involving cyclic variables. The governing logic is: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. Step-by-Step Solution: To simplify the messy expression, assign variables to each unique bracketed difference: • Let a = 4.53 - 3.07 • Let b = 3.07 - 2.15 • Let c = 2.15 - 4.53 First, test the condition by adding the variables together: a + b + c = (4.53 - 3.07) + (3.07 - 2.15) + (2.15 - 4.53) a + b + c = 4.53 - 4.53 + 3.07 - 3.07 + 2.15 - 2.15 = 0 Because they sum to zero, the rule a^3 + b^3 + c^3 = 3abc is actively unlocked. Next, rewrite the original fraction expression using our variables: (a^2/bc) + (b^2/ca) + (c^2/ab) To add these fractions, find a common denominator, which is abc. Multiply the numerator and denominator of each fraction by the missing variable: (a × a^2/abc) + (b × b^2/abc) + (c × c^2/abc) (a^3 + b^3 + c^3/abc) Now, substitute the active rule (a^3 + b^3 + c^3 = 3abc) into the numerator: (3abc/abc) Cancel out the common term abc from the top and bottom. The final result is simply 3. Exam Strategy & Shortcut: Whenever you see a cyclic pattern of differences—where numbers chain together like (x-y), (y-z), (z-x)—they are guaranteed to sum to zero. If you encounter an expression formatted as squares over products of these cyclic differences (e.g., a^2/bc + b^2/ca + c^2/ab), bypass all algebra. The mathematical structure dictates the answer will always reduce to exactly 3. Common Pitfall: Attempting to perform the manual arithmetic. Calculating (4.53 - 3.07)^2 by hand will consume an enormous amount of time and almost guarantee an arithmetic error. This question is a pure test of algebraic pattern recognition, not calculation speed. Final Answer: Therefore, the correct answer is 3.

Question 4

Solve $\frac{3}{5}$ of $\frac{4}{7}$ of $\frac{5}{12}$ of $1015 = x$

Accepted Answer

Concept & Formula The word "of" in mathematical word problems and simplification questions translates directly to multiplication. The strategy is to multiply all the fractions together first and cancel out common terms in the numerators and denominators before multiplying by the large whole number. Step-by-Step Solution • Write out the full equation replacing "of" with multiplication signs: x = (3/5) × (4/7) × (5/12) × 1015 • Isolate the fraction part to simplify it first: ( (3 × 4 × 5/5 × 7 × 12) ) • Notice that the 5 in the numerator cancels with the 5 in the denominator. • Notice that 3 × 4 = 12 in the numerator, which perfectly cancels out the 12 in the denominator. • The entire fraction expression simplifies to: (1/7) • Now multiply this simplified fraction by the whole number: x = (1/7) × 1015 • Perform the final division: 1015 \div 7 = 145 Exam Strategy & Shortcut Never multiply the first fraction by the large number immediately. Always scan the chain of fractions for cross-cancellation. In this problem, visually spotting that (3 × 4) cancels 12, and 5 cancels 5, reduces the entire problem to a single step: 1015 \div 7, taking mere seconds. Common Pitfall A common time-wasting mistake is trying to solve the chain sequentially from left to right (e.g., finding (3/5) of something, then (4/7) of that result). This creates messy intermediate decimals or large irreducible fractions. Final Answer Therefore, the correct answer is 145.

Question 5

Solve $1\frac{1}{8} + 1\frac{6}{7} + 3\frac{3}{5} = x$

Accepted Answer

Concept & Strategy When adding multiple mixed fractions, the most efficient method is to split the numbers into their integer components and their fractional components. Process them separately before recombining. A(b/c) + X(y/z) = (A + X) + ((b/c) + (y/z)) Step-by-Step Solution • Separate the integer and fraction parts from the given equation: (1 + 1 + 3) + ((1/8) + (6/7) + (3/5)) • Sum the integer parts: 1 + 1 + 3 = 5 • Find the Least Common Multiple (LCM) for the denominators 8, 7, and 5 to add the fractions. Since they share no common factors, multiply them: LCM = 8 × 7 × 5 = 280 • Convert each fraction to have the common denominator of 280: (1 × 35/8 × 35) = (35/280) (6 × 40/7 × 40) = (240/280) (3 × 56/5 × 56) = (168/280) • Add the numerators of the converted fractions together: (35 + 240 + 168/280) = (443/280) • Convert the improper fraction back into a mixed fraction: (443/280) = 1(163/280) • Recombine the integer sum and the new mixed fraction: 5 + 1(163/280) = 6(163/280) Exam Strategy & Shortcut Instead of immediately jumping to find the massive LCM of 280, add the whole numbers first to get 5. Glancing at the fractions (6/7) (which is nearly 1) and (3/5) (more than half), you know their sum will add at least 1 more to your total integer. This means the final integer part must be 6. Looking at the options, (b) is the only choice starting with 6, allowing you to bypass the heavy fraction math entirely. Common Pitfall A frequent error is converting all mixed fractions into improper fractions right at the start (e.g., turning 3(3/5) into (18/5)). This results in very large, unwieldy numerators that must be multiplied for the common denominator, drastically increasing calculation time and the likelihood of arithmetic mistakes. Final Answer Therefore, the correct answer is 6(163/280).

Question 6

Solve $323.46 + 23.04 - 43.17 - x = 303$

Accepted Answer

Concept & Strategy This is a straightforward decimal arithmetic problem. The most effective approach is to follow the order of operations by grouping and solving the additions first, then the subtractions, and finally isolating the unknown variable using basic algebraic principles. Step-by-Step Solution • Group the positive numbers on the left side of the equation and sum them: 323.46 + 23.04 = 346.50 • Substitute this sum back into the equation: 346.50 - 43.17 - x = 303 • Perform the subtraction on the left side: 346.50 - 43.17 = 303.33 • Now the equation is simplified to: 303.33 - x = 303 • Rearrange the equation to solve for x by moving x to the right side and 303 to the left side: 303.33 - 303 = x • Calculate the final difference: x = 0.33 Exam Strategy & Shortcut Use the Unit Digit or "Decimal Digit" method to bypass full calculations. Look only at the hundredths place of the numbers: 0.06 + 0.04 - 0.07 = 0.10 - 0.07 = 0.03. This means the final answer for x must end in a 3. Since the right side of the equation is a whole integer (303.00), the decimal portion of x must be exactly 0.33 to balance the 303.33 on the left. This immediately points to option (a) without doing full vertical math. Common Pitfall A common mistake is misaligning the decimal points when doing vertical addition or subtraction, especially when numbers have different amounts of digits before the decimal. Always ensure the decimal points are perfectly stacked. Final Answer Therefore, the correct answer is 0.33.

Question 7

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 8

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 9

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 10

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 11

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 12

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 13

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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