An arithmetic expression is often made difficult by its presentation rather than its underlying calculation. A fraction may hide a cancellation; a recurring decimal may look longer than it is; a mixed expression may tempt a candidate to calculate from left to right without regard to the order of operations. The aim of this chapter is to make each expression smaller before making it numerical. A reliable solution has three stages: interpret the notation, choose an efficient representation and check that the result is of a reasonable size.
1. Fractions describe a relationship
The fraction a/b means a divided by b, with b not equal to zero. The numerator counts the parts being used; the denominator states the size of each part relative to a whole. A fraction can also be a ratio, a point on the number line or the result of a division. These meanings agree mathematically, but the context determines the best method. In a sharing problem, a/b of a quantity Q means (a × Q)/b. In an expression, a/b may be kept as a single exact number until the final step.
Equivalent fractions are formed by multiplying or dividing numerator and denominator by the same non-zero number. Thus 18/24 = 3/4 after division by 6. This is a simplification, not a change in value. A common error is to cancel terms across addition, as if (a+b)/b could be reduced to a. In fact, (a+b)/b = a/b + 1. Cancellation applies to factors in a product, not to terms joined by a plus or minus sign.
The value of a fraction increases when its positive numerator increases while its denominator stays fixed. It decreases when its positive denominator increases while its numerator stays fixed. When both change, neither shortcut alone is reliable. Compare 7/12 and 5/8 by cross multiplication: 7 × 8 = 56 while 5 × 12 = 60, so 7/12 < 5/8. Both denominators are positive, which keeps the inequality direction unchanged.
Worked example 1. Arrange 5/6, 7/9 and 11/12 in ascending order. Use denominator 36: 5/6 = 30/36, 7/9 = 28/36 and 11/12 = 33/36. Therefore 7/9 < 5/6 < 11/12. A quick check using decimals, about 0.78, 0.83 and 0.92, confirms the order without replacing the exact proof.
2. Addition, subtraction, multiplication and division
For addition or subtraction, use a common denominator. The least common multiple of the denominators minimizes arithmetic, but any valid common multiple works. For example, 2/15 + 7/10 uses denominator 30: 4/30 + 21/30 = 25/30 = 5/6. Do not add denominators: 2/15 + 7/10 is not 9/25.
For multiplication, multiply numerators and denominators, cancelling common factors before multiplying when possible. For division, multiply by the reciprocal of a non-zero divisor. It is helpful to say the operation aloud: “divide by 3/5” becomes “multiply by 5/3”. This habit prevents the common mistake of inverting the first fraction.
Worked example 2. Calculate (14/25) × (15/28). Cancel 14 against 28 to get 1/2, and 15 against 25 to get 3/5. The product is (1 × 3)/(5 × 2) = 3/10. Multiplying first would give 210/700 and then the same answer, but creates larger numbers and more room for error.
Worked example 3. Calculate 7/12 ÷ 14/9. Replace division by multiplication: (7/12) × (9/14). Cancel 7 against 14, giving 1 and 2; cancel 9 against 12, giving 3 and 4. The answer is 3/8. Check its size: dividing a number below 1 by a number above 1 should make it smaller, so 3/8 is plausible.
When an expression contains an integer next to a fraction, treat the integer as a fraction over 1. A mixed number such as 2 1/3 means 2 + 1/3 = 7/3. It does not mean 2 × 1/3. In multiplication and division, convert mixed numbers to improper fractions before applying rules. When the question asks for the final answer as a mixed number, convert back only at the end.
3. Fractions of a quantity and the changing base
The phrase “one-third of the remainder” changes the base. If a person spends 1/4 of ₹1,200, the remainder is ₹900. One-third of the remainder is ₹300, not ₹400. In multistep word problems, record the current base after each operation. A compact table with “before”, “fraction used” and “after” is safer than trying to apply all percentages or fractions to the original amount.
Worked example 4. A candidate completes 2/5 of a book on Monday and 1/3 of the remaining pages on Tuesday. What fraction of the whole book remains? After Monday, 3/5 remains. Tuesday's reading is (1/3) × (3/5) = 1/5 of the whole book. The final remainder is 3/5 − 1/5 = 2/5. An alternative route multiplies the successive remaining fractions: (3/5) × (2/3) = 2/5.
If 3/8 of a number is 27, the whole is 27 ÷ (3/8) = 27 × 8/3 = 72. The inverse operation is essential: multiplying 27 by 3/8 would find 3/8 of 27, a different question. Estimate: since 3/8 is less than one half, the whole must be more than twice 27, which 72 is.
4. Finite and recurring decimals
A decimal with finitely many digits is rational. Write it over a power of ten and reduce: 0.375 = 375/1000 = 3/8. The number of digits after the point determines the initial denominator. For a fraction in lowest terms, its decimal expansion terminates if the denominator has no prime factors other than 2 and 5. This follows because a power of ten contains only those two primes. Thus 7/40 terminates, while 7/12 recurs because 12 has a factor 3 after reduction.
A recurring decimal is also rational. For x = 0.272727…, multiply by 100 because the repeating block has two digits: 100x = 27.272727…. Subtract x to obtain 99x = 27, hence x = 27/99 = 3/11. The subtraction removes the infinite repeated tail; it is exact, not an approximation. If a non-recurring prefix appears before the repeating block, align both parts before subtracting.
Worked example 5. Convert 0.1666… to a fraction, where only 6 repeats. Let x = 0.1666…. Then 10x = 1.6666… and 100x = 16.6666…. Subtract 10x from 100x: 90x = 15, so x = 1/6. A rough check is 1/6 ≈ 0.17.
Decimal place value matters in multiplication and division. For 0.24 × 0.05, multiply 24 × 5 = 120 and count four decimal places in total: 0.0120 = 0.012. For 1.2 ÷ 0.03, move the decimal point two places in both dividend and divisor to get 120 ÷ 3 = 40. The same scaling must be applied to both quantities; otherwise the quotient changes.
5. BODMAS is a grouping rule, not a race
Resolve brackets first, then powers or roots, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not always precede division; they share a level. Likewise, addition does not always precede subtraction. The notation 18 ÷ 3 × 2 evaluates as (18 ÷ 3) × 2 = 12. It is not 18 ÷ (3 × 2) unless brackets explicitly make 3 × 2 the divisor.
A horizontal fraction bar groups its entire numerator and its entire denominator. The expression (8 + 4)/(6 − 2) equals 12/4 = 3. Removing the grouping and writing 8 + 4 ÷ 6 − 2 changes the expression. In handwritten work, draw the fraction bar long enough to show what belongs above and below it.
Worked example 6. Evaluate 3/4 + 2 × (5/6 − 1/3). The bracket gives 5/6 − 2/6 = 1/2. Next, 2 × 1/2 = 1. Finally, 3/4 + 1 = 7/4. The expression is greater than 1, so a result such as 7/12 would signal an order-of-operations mistake.
The word “of” in exam arithmetic usually denotes multiplication. Thus 1/2 of 3/4 is (1/2) × (3/4) = 3/8. When “of” appears alongside division, rewrite the entire line with explicit multiplication signs and brackets before calculating. Poor typography can make such items ambiguous; a good exam book must print the intended grouping clearly rather than teaching a special exception to normal arithmetic rules.
6. Approximation and error control
An exact answer is preferred when the options are close or when subsequent steps depend on it. Approximation is useful for eliminating implausible choices and checking magnitude. Round only after understanding which place value is relevant. For example, 19.8 × 5.1 is near 20 × 5 = 100, so an answer near 10 or 1,000 is plainly wrong. The exact product is 100.98.
Rounding at intermediate steps can accumulate error. If the question asks for a final amount to two decimal places, keep enough digits during calculation and round once at the end. A fraction such as 1/3 should be held as 1/3 rather than replaced by 0.33 if the exact fraction will later cancel. Where an approximation is requested, state the rounding rule and unit in the final answer.
Worked example 7. Estimate 49.7 × 20.4 ÷ 10.1 before computing. Use 50 × 20 ÷ 10 = 100. The exact value is a little above 100 because the numerator product is 1,013.88 and the denominator is 10.1; 1,013.88 ÷ 10.1 ≈ 100.38. A reported value of about 10 reveals a misplaced decimal.
Check signs as well as magnitude. A negative divided by a negative is positive. Subtracting a larger positive fraction from a smaller one produces a negative answer. If a physical word problem asks for length or time, a negative computed result usually means the equation or order of subtraction was set up incorrectly.
7. A compact exam method
Before solving, identify whether the expression is an exact-fraction, decimal-conversion, successive-base or approximation problem. Copy the grouping carefully. Simplify by cancellation before multiplying, and use the least common denominator for addition. At the end, compare the answer's sign and size with the original question. This five-second check is often more valuable than a second full calculation.
An error log should distinguish concept errors from arithmetic slips. “Added denominators” points to a concept error; “3 × 8 written as 21” is an arithmetic slip; “used the original total after a remainder step” is a base-selection error. Different errors require different practice. Retrying the same question without naming the mistake can create a false sense of mastery.
Worked example 8. Evaluate (2 1/4 − 5/6) ÷ (1/3). Convert 2 1/4 to 9/4. The bracket becomes 9/4 − 5/6 = 27/12 − 10/12 = 17/12. Dividing by 1/3 gives (17/12) × 3 = 17/4 = 4 1/4. Check: the bracket is about 2.25 − 0.83 = 1.42; dividing by one-third should roughly triple it, giving about 4.25.
Recall before practice
1. Explain why cancellation is valid in (6 × 5)/(9 × 5) but invalid in (6 + 5)/(9 + 5). 2. State the condition for a reduced fraction to have a terminating decimal. 3. Explain why “one-third of the remainder” uses a different base from “one-third of the original”. 4. Evaluate 24 ÷ 4 × 3 and name the rule that fixes its order. 5. Convert 0.272727… to a fraction without treating the dots as a finite string of digits.
Chapter 3 practice — 50 questions
Choose one option for each item. Keep a separate answer list; explanations follow this set.
1. Find the value of 198 ÷ 3 of 6 + 4
A. 400 B. 70 C. 7 D. 15
2. What will come in place of the question mark (?) in the following equation? √144 × √49 + ? = 29 × 12
A. 329 B. 264 C. 432 D. 528
3. Simplify: 1/6 − 3 1/2 − 2 2/3
A. -4 B. -6 C. 6 1/3 D. -2/3
4. How many digits will there be after the decimal point in the product 0.3 × 0.81?
A. 3 B. 4 C. 2 D. 5
5. Express 1/20 as a decimal.
A. 0.05 B. 0.1 C. 0.5 D. 0.005
6. Which of the following is the largest?
A. 0.09 B. 0.909 C. 0.098 D. 0.989
7. If (m + 3) × 3 + 11 − 11 = 27, then the value of m is:
A. 6 B. 7 C. 9 D. 5
8. How many digits will there be after the decimal point in the product 0.4845 × 0.909?
A. 7 B. 4 C. 8 D. 6
9. If 26 × 26 = 676, then the value of 0.26 × 26 is:
A. 67.6 B. 0.676 C. 6.76 D. 0.0676
10. What will come in place of the question mark (?) in the following equation? √4 × √256 + ? = 11 × 3
A. 65 B. 2 C. 15 D. 1
11. How many digits will there be after the decimal point in the product 2.65 × 1.47?
A. 3 B. 4 C. 5 D. 2
12. What will come in place of the question mark (?) in the following equation? 80% of 850 + 90% of 650 = ?
A. 7385 B. 95 C. 1265 D. 6530
13. Which of the following is the smallest?
A. 0.11 B. 0.01 C. 0.011 D. 0.181
14. What will come in place of the question mark (?) in the following equation? 60 × 4 + 35 ÷ 7 − √? × 4 = 149
A. 529 B. 576 C. 625 D. 24
15. What is the 18th digit after the decimal point in the decimal expansion of 7/13?
A. 1 B. 5 C. 2 D. 6
16. Simplify: (2.5 × 2.5 × 2.5 + 0.5 × 0.5 × 0.5) ÷ (2.5 × 2.5 − 2.5 × 0.5 + 0.5 × 0.5)
A. 2 B. 3 C. 1.5 D. 1.25
17. Find the value of 0.135 ÷ 0.009.
A. 1.5 B. 150 C. 15 D. 0.15
18. If A = 754.24/32, B = 464.13/81, C = 838.08/24, arrange A, B and C in descending order.
A. C > A > B B. A > C > B C. B > A > C D. B > C > A
19. The value of the following expression is: 90 ÷ 6 of 5 + 5 × 3 − 4
A. 14 B. -2 C. 22 D. 86
20. Simplify: 89.287 + 299.6 + 255.85 − 266.57 − 59
A. 970.307 B. 1122.75 C. 437.167 D. 319.167
21. If m × 6 + 9 − 12 ÷ 2 − 7 + 2 ÷ 2 × 4 + 8 = 38, then the value of m is:
A. 1 B. 4 C. 6 D. 5
22. Express 0.838383… (the block 83 recurring) as a fraction in its lowest terms.
A. 83/100 B. 83/99 C. 83/999 D. 83/90
23. Simplify: 1 + 1/(5 + 1/6)
A. 31/37 B. 1 6/31 C. 1 1/5 D. 6 1/6
24. If 322 ÷ 23 = 14, then the value of 3.22 ÷ 0.23 is:
A. 14 B. 0.0014 C. 140000 D. 1.4
25. Find the value of 14 − [37 − {39 ÷ 3 − (8 − 3 ÷ 3)} × 4]
A. -7 B. -5 C. 1 D. -110
26. Find the value of 2 1/2 ÷ 1/5 of 5/7 + 4/9
A. 17 17/18 B. 17 1/18 C. 10 115/146 D. 9 47/126
27. Simplify: 19 − [17 − {6 ÷ 2 − (5 − 9 ÷ 3)} × 3]
A. -13 B. 5 C. 4 D. -29
28. Find the value of 10% of 7.5 + 7.5 ÷ 2.5 + 3.5 × 4
A. 17.75 B. 14.75 C. 20.75 D. 33.5
29. Express 0.141414… (the block 14 recurring) as a fraction in its lowest terms.
A. 14/99 B. 14/999 C. 7/50 D. 7/45
30. Find the value of 38 − [34 − {24 ÷ 2 − (12 − 30 ÷ 5)} × 5]
A. 28 B. -26 C. 34 D. -102
31. If 57/0.57 = 960/x, then the value of x is:
A. 0.96 B. 96 C. 96000 D. 9.6
32. Find the value of 8007 − (5868 ÷ 586.8).
A. 7997 B. 8017 C. 8006 D. 7907
33. The value of the following expression is: 120 ÷ 4 of 6 + 9 × 6 − 53
A. 181 B. 6 C. -418 D. 112
34. If 62 × 62 = 3844, then the value of 0.62 × 6.2 is:
A. 3.844 B. 38.44 C. 0.03844 D. 0.3844
35. If 49 × 43 = 2107, then the value of 0.49 × 0.43 is:
A. 0.2107 B. 21.07 C. 0.02107 D. 2.107
36. Find the value of 7332 − (2244 ÷ 2.244).
A. 8332 B. 6342 C. 6332 D. 7232
37. What is the 12th digit after the decimal point in the decimal expansion of 5/9?
A. 8 B. 6 C. 4 D. 5
38. The value of the following expression is: 25 − [19 − {36 ÷ 4 − (8 − 18 ÷ 6)} × 2]
A. 10 B. -5 C. 2 D. 14
39. Find the value of 75% of 1.5 + 27.5 ÷ 2.5 + 3.5 × 15
A. 75.625 B. 64.625 C. 53.625 D. 122.375
40. Simplify: 1 2/3 − 4 1/3 + 5/6 − 3 3/5
A. 10 13/30 B. 1 23/30 C. -6 17/30 D. -5 13/30
41. If 5 + 1/(4 + 1/x) = 68/13, then the value of x is:
A. 2 B. 3 C. 9 D. 4
42. The value of 0.555… − 0.444… expressed as a fraction is:
A. 1/18 B. 1/10 C. 5/9 D. 1/9
43. If 0.25 × x = 0.5 × 3.2, then x is equal to:
A. 6.4 B. 64 C. 0.0390625 D. 0.64
44. If 5 + 1/(3 + 1/x) = 85/16, then the value of x is:
A. 5 B. 4 C. 3 D. 6
45. If 0.4 × x = 0.75 × 0.8, then x is equal to:
A. 1.5 B. 0.15 C. 15 D. 0.375
46. Simplify: 2.4 × 4 + 3.6 ÷ 0.8 of 0.5 − 1.25 + 0.2 of 7.5
A. 21.35 B. 12.1 C. 18.85 D. 17.35
47. If A = 1404.48/96, B = 933/75, C = 655.65/45, arrange A, B and C in ascending order.
A. B < C < A B. C < A < B C. C < B < A D. A < B < C
48. Simplify: (0.45 × 0.35) ÷ 0.15 + 3.5
A. 4.55 B. 5.55 C. 14 D. 3.605
49. What will come in place of the question mark (?) in the following equation? 3^? × 13 = 3^3 × 3^4 × 13
A. 12 B. 8 C. 6 D. 7
50. Simplify: (3 × 3 + 2 × 3 × 1.5 + 1.5 × 1.5) ÷ (3 × 3 − 1.5 × 1.5)
A. 4 B. 3 C. 4.5 D. 6.75
Chapter 3 — Answers and explanations
After checking the key, re-solve any miss without looking at the formula.
1. D. 'of' is done before division: 3 of 6 = 18. Then 198 ÷ 18 = 11, and 11 + 4 = 15. (Dividing by 3 first and then multiplying by 6 gives 400, which is wrong.)
APAR26-03-31 | BODMAS with the 'of' operator | Easy
2. B. √144 = 12 and √49 = 7, so the left side is 12 × 7 + ? = 84 + ?. Right side: 29 × 12 = 348. Hence ? = 348 − 84 = 264.
APAR26-03-26 | Missing value (?) equation | Easy
3. B. Separate whole and fractional parts. Whole parts: 0 − 3 − 2 = -5. Fractional parts with LCM 6: 1/6 −3/6 −4/6 = -1. Total = -5 + (-1) = -6 = -6.
APAR26-03-32 | Mixed fraction arithmetic | Easy
4. A. Decimal places: 0.3 → 1, 0.81 → 2; sum = 3. Multiplying the digits without the points: 3 × 81 = 243, which does not end in 0, so all 3 places survive: product = 0.243.
APAR26-03-03 | Decimal places in a product | Easy
5. A. multiply numerator and denominator by 5 to make the denominator a power of 10: 1/20 = 5/100 = 0.05. Hence 1/20 = 0.05.
APAR26-03-07 | Fraction to decimal | Easy
6. D. Write all four as decimals with the same number of places: 0.09 = 0.09, 0.909 = 0.909, 0.098 = 0.098, 0.989 = 0.989. In decreasing order: 0.989 > 0.909 > 0.098 > 0.09. The largest is 0.989.
APAR26-03-01 | Comparing decimals and fractions | Easy
7. A. Move the constants: (m + 3) × 3 = 27 − 11 + 11 = 27. So m + 3 = 27/3 = 9, giving m = 9 − 3 = 6.
APAR26-03-27 | Unknown in a BODMAS expression | Easy
8. A. Decimal places: 0.4845 → 4, 0.909 → 3; sum = 7. Multiplying the digits without the points: 4845 × 909 = 4404105, which does not end in 0, so all 7 places survive: product = 0.4404105.
APAR26-03-06 | Decimal places in a product | Easy
9. C. Ignore the decimal points: 26 × 26 = 676. Count the decimal places in the factors: 0.26 has 2, 26 has 0, total 2. Put the point 2 places from the right in 676: 6.76.
APAR26-03-02 | Product with shifted decimals | Easy
10. D. √4 = 2 and √256 = 16, so the left side is 2 × 16 + ? = 32 + ?. Right side: 11 × 3 = 33. Hence ? = 33 − 32 = 1.
APAR26-03-28 | Missing value (?) equation | Easy
11. B. Decimal places: 2.65 → 2, 1.47 → 2; sum = 4. Multiplying the digits without the points: 265 × 147 = 38955, which does not end in 0, so all 4 places survive: product = 3.8955.
APAR26-03-05 | Decimal places in a product | Easy
12. C. 80% of 850 = 80 × 850/100 = 680; 90% of 650 = 90 × 650/100 = 585. Adding, ? = 680 + 585 = 1265.
APAR26-03-30 | Percent-of equation with a missing term | Easy
13. B. Write all four as decimals with the same number of places: 0.11 = 0.11, 0.01 = 0.01, 0.011 = 0.011, 0.181 = 0.181. In decreasing order: 0.181 > 0.11 > 0.011 > 0.01. The smallest is 0.01.
APAR26-03-04 | Comparing decimals and fractions | Easy
14. B. 60 × 4 = 240 and 35 ÷ 7 = 5, so 240 + 5 − √? × 4 = 149 ⇒ √? × 4 = 96 ⇒ √? = 24 ⇒ ? = 24² = 576. (Marking 24 itself is the trap: the question asks for ?, not √?.)
APAR26-03-29 | Missing value (?) equation | Easy
15. A. 7/13 = 0.(538461)(538461)… — the block "538461" of 6 digits repeats. Position 18 within the cycle: 18 = 6 × 2 + 6, so it is the 6th digit of the block, i.e. 1.
APAR26-03-15 | nth digit of a recurring decimal | Medium
16. B. Use a³ + b³ = (a + b)(a² − ab + b²). The denominator is exactly a² − ab + b², so it cancels and the value is a + b = 2.5 + 0.5 = 3.
APAR26-03-40 | Algebraic identity shortcut | Medium
17. C. Make the divisor a whole number: multiply both by 1000: 0.135 ÷ 0.009 = 135 ÷ 9. Now 135 ÷ 9 = 15. (Shifting only one of the two numbers gives 1.5 or 150.)
APAR26-03-11 | Decimal divided by decimal | Medium
18. A. Work out each quotient: A = 754.24 ÷ 32 = 23.57; B = 464.13 ÷ 81 = 5.73; C = 838.08 ÷ 24 = 34.92. Largest to smallest: C (34.92) > A (23.57) > B (5.73), i.e. C > A > B.
APAR26-03-16 | Ordering decimal quotients | Medium
19. A. 'of' first: 6 of 5 = 30, so 90 ÷ 30 = 3. Multiplication: 5 × 3 = 15. Then 3 + 15 − 4 = 14. (Treating 90 ÷ 6 × 5 left to right gives 86, the standard trap.)
APAR26-03-42 | BODMAS with the 'of' operator | Medium
20. D. Line up the decimal points (write every term to 3 decimal places). Positives: 89.287 + 299.6 + 255.85 = 644.737; negatives: 266.57 + 59 = 325.57. Result = 644.737 − 325.57 = 319.167.
APAR26-03-19 | Adding and subtracting decimals | Medium
21. D. Evaluate the constants first: 12 ÷ 2 = 6; 2 ÷ 2 × 4 = 1 × 4 = 4. Constants total = 9 − 6 − 7 + 4 + 8 = 8. So 6m + (8) = 38 ⇒ 6m = 30 ⇒ m = 5.
APAR26-03-34 | Unknown in a BODMAS expression | Medium
22. B. Let x = 0.838383…. Multiply by 10^2: 100x = 83.8383… Subtract: 100x − x = 83 ⇒ 99x = 83 ⇒ x = 83/99 = 83/99. (Rule: a pure recurring block of 2 digits goes over 99; 83/100 would be the terminating decimal 0.83.)
APAR26-03-10 | Pure recurring decimal to fraction | Medium
23. B. Work from the innermost fraction outward: 5 + 1/(6) = 5 + 1/6 = 31/6; then 1 + 1/(31/6) = 1 + 6/31 = 37/31. Hence the value is 37/31 = 1 6/31.
APAR26-03-39 | Continued fraction | Medium
24. A. 3.22 = 322 × 10^−2 and 0.23 = 23 × 10^−2. So 3.22 ÷ 0.23 = (322 ÷ 23) × 10^(2 − 2) = 14 × 10^0 = 14. (In division the decimal places subtract, they do not add — adding them gives 0.0014.)
APAR26-03-20 | Quotient with shifted decimals | Medium
25. C. Innermost bracket: (8 − 3 ÷ 3) = 8 − 1 = 7. Curly bracket: {39 ÷ 3 − 7} = 13 − 7 = 6. Square bracket: [37 − 6 × 4] = 37 − 24 = 13. Finally 14 − 13 = 1.
APAR26-03-33 | BODMAS with brackets (integers) | Medium
26. A. Convert to improper fractions: 2 1/2 = 5/2, 1/5 = 1/5, 5/7 = 5/7, 4/9 = 4/9. 'of' first: 1/5 × 5/7 = 1/7. Then 5/2 ÷ 1/7 = 35/2. Finally 35/2 + 4/9 = 323/18 = 17 17/18.
APAR26-03-37 | BODMAS with mixed fractions | Medium
27. B. Innermost bracket: (5 − 9 ÷ 3) = 5 − 3 = 2. Curly bracket: {6 ÷ 2 − 2} = 3 − 2 = 1. Square bracket: [17 − 1 × 3] = 17 − 3 = 14. Finally 19 − 14 = 5.
APAR26-03-35 | BODMAS with brackets (integers) | Medium
28. A. Evaluate each term separately: 10% of 7.5 = 0.75; 7.5 ÷ 2.5 = 3; 3.5 × 4 = 14. Adding all the terms: 0.75 + 3 + 14 = 17.75.
APAR26-03-43 | Decimals, percentages and 'of' | Medium
29. A. Let x = 0.141414…. Multiply by 10^2: 100x = 14.1414… Subtract: 100x − x = 14 ⇒ 99x = 14 ⇒ x = 14/99 = 14/99. (Rule: a pure recurring block of 2 digits goes over 99; 14/100 would be the terminating decimal 0.14.)
APAR26-03-08 | Pure recurring decimal to fraction | Medium
30. C. Innermost bracket: (12 − 30 ÷ 5) = 12 − 6 = 6. Curly bracket: {24 ÷ 2 − 6} = 12 − 6 = 6. Square bracket: [34 − 6 × 5] = 34 − 30 = 4. Finally 38 − 4 = 34.
APAR26-03-36 | BODMAS with brackets (integers) | Medium
31. D. 57/0.57: move the decimal 2 places in both to get 5700/57 = 100. So 100 = 960/x ⇒ x = 960/100 = 9.6. (Each division by 10 shifts the point one place left; miscounting the shift gives 96 or 0.96.)
APAR26-03-14 | Proportion with decimals | Medium
32. A. 586.8 is 5868 with the decimal point moved 1 place left, i.e. 586.8 = 5868/10. So 5868 ÷ 586.8 = 5868 × 10/5868 = 10. Then 8007 − 10 = 7997.
APAR26-03-12 | Division by a place-shifted decimal | Medium
33. B. 'of' first: 4 of 6 = 24, so 120 ÷ 24 = 5. Multiplication: 9 × 6 = 54. Then 5 + 54 − 53 = 6. (Treating 120 ÷ 4 × 6 left to right gives 181, the standard trap.)
APAR26-03-45 | BODMAS with the 'of' operator | Medium
34. A. Ignore the decimal points: 62 × 62 = 3844. Count the decimal places in the factors: 0.62 has 2, 6.2 has 1, total 3. Put the point 3 places from the right in 3844: 3.844.
APAR26-03-09 | Product with shifted decimals | Medium
35. A. Ignore the decimal points: 49 × 43 = 2107. Count the decimal places in the factors: 0.49 has 2, 0.43 has 2, total 4. Put the point 4 places from the right in 2107: 0.2107.
APAR26-03-18 | Product with shifted decimals | Medium
36. C. 2.244 is 2244 with the decimal point moved 3 places left, i.e. 2.244 = 2244/1000. So 2244 ÷ 2.244 = 2244 × 1000/2244 = 1000. Then 7332 − 1000 = 6332.
APAR26-03-13 | Division by a place-shifted decimal | Medium
37. D. 5/9 = 0.(5)(5)… — the block "5" of 1 digit repeats. Position 12 within the cycle: 12 = 1 × 11 + 1, so it is the 1st digit of the block, i.e. 5.
APAR26-03-17 | nth digit of a recurring decimal | Medium
38. D. Innermost bracket: (8 − 18 ÷ 6) = 8 − 3 = 5. Curly bracket: {36 ÷ 4 − 5} = 9 − 5 = 4. Square bracket: [19 − 4 × 2] = 19 − 8 = 11. Finally 25 − 11 = 14.
APAR26-03-41 | BODMAS with brackets (integers) | Medium
39. B. Evaluate each term separately: 75% of 1.5 = 1.125; 27.5 ÷ 2.5 = 11; 3.5 × 15 = 52.5. Adding all the terms: 1.125 + 11 + 52.5 = 64.625.
APAR26-03-44 | Decimals, percentages and 'of' | Medium
40. D. Separate whole and fractional parts. Whole parts: 1 − 4 + 0 − 3 = -6. Fractional parts with LCM 30: 20/30 −10/30 +25/30 −18/30 = 17/30. Total = -6 + (17/30) = -163/30 = -5 13/30.
APAR26-03-38 | Mixed fraction arithmetic | Medium
41. B. Peel the layers: 1/(4 + 1/x) = 68/13 − 5 = 3/13 ⇒ 4 + 1/x = 13/3 ⇒ 1/x = 13/3 − 4 = 1/3 ⇒ x = 3.
APAR26-03-47 | Continued fraction (unknown) | Difficult
42. D. A single recurring digit d gives d/9: 0.555… = 5/9 and 0.444… = 4/9. So the value = 5/9 − 4/9 = 1/9. (Using 10 in the denominator, as for 0.5, is the trap.)
APAR26-03-22 | Operations on recurring decimals | Difficult
43. A. Right side: 0.5 × 3.2 = 1.6 (1 + 1 = 2 decimal places before dropping trailing zeros). So x = 1.6 ÷ 0.25 = 160 ÷ 25 = 6.4. (Multiply numerator and denominator by 100 to clear the decimal in the divisor.)
APAR26-03-25 | Decimal equation | Difficult
44. A. Peel the layers: 1/(3 + 1/x) = 85/16 − 5 = 5/16 ⇒ 3 + 1/x = 16/5 ⇒ 1/x = 16/5 − 3 = 1/5 ⇒ x = 5.
APAR26-03-46 | Continued fraction (unknown) | Difficult
45. A. Right side: 0.75 × 0.8 = 0.6 (2 + 1 = 3 decimal places before dropping trailing zeros). So x = 0.6 ÷ 0.4 = 6 ÷ 4 = 1.5. (Multiply numerator and denominator by 10 to clear the decimal in the divisor.)
APAR26-03-24 | Decimal equation | Difficult
46. C. Term by term: 2.4 × 4 = 9.6; 0.8 of 0.5 = 0.4, so 3.6 ÷ 0.4 = 9; 0.2 of 7.5 = 1.5. Now 9.6 + 9 − 1.25 + 1.5 = 18.85. ('of' binds before ÷; doing 3.6 ÷ 0.8 first is the trap.)
APAR26-03-48 | Decimals, percentages and 'of' | Difficult
47. A. Work out each quotient: A = 1404.48 ÷ 96 = 14.63; B = 933 ÷ 75 = 12.44; C = 655.65 ÷ 45 = 14.57. Smallest to largest: B (12.44) < C (14.57) < A (14.63), i.e. B < C < A.
APAR26-03-21 | Ordering decimal quotients | Difficult
48. A. Bracket first: 0.45 × 0.35 = 0.1575 (2 + 2 decimal places). Then 0.1575 ÷ 0.15: multiply both by 100 → 15.75 ÷ 15 = 1.05. Finally 1.05 + 3.5 = 4.55.
APAR26-03-23 | Decimal chain with division | Difficult
49. D. Divide both sides by 13: 3^? = 3^3 × 3^4. Same base, so add the indices: 3^? = 3^7 (= 2187). Hence ? = 7. (Multiplying the indices, 3 × 4 = 12, is the trap.)
APAR26-03-50 | Missing value (?) equation | Difficult
50. B. Numerator = (a + b)²; denominator = (a + b)(a − b). Cancel one (a + b): value = (a + b)/(a − b) = (3 + 1.5)/(3 − 1.5) = 4.5/1.5 = 3.
APAR26-03-49 | Algebraic identity shortcut | Difficult