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Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 4

Arithmetic: BODMAS, Ratio and Proportion, Percentage

What to remember

  • BODMAS fixes the order: Brackets, Orders (powers and roots), Division and Multiplication (left to right), then Addition and Subtraction (left to right); division and multiplication have equal rank, and so do addition and subtraction.
  • A ratio compares two quantities of the same kind and has no unit; a proportion says two ratios are equal, so the product of extremes equals the product of means (a : b = c : d gives ad = bc).
  • A percentage is a fraction with denominator 100; the base of the percentage is always the quantity that comes after "of" or "than", and a rise followed by an equal fall never returns to the start.

BODMAS (order of operations)

BODMAS tells us which operation to do first so that everyone gets the same answer. It is also called PEMDAS or BIDMAS in other books.

  • 1. Brackets: do the work inside brackets first.
  • 2. Orders: powers (squares, cubes) and roots. "Of" is also done at this stage, as in "half of 40".
  • 3. D and M: division and multiplication, moving from left to right.
  • 4. A and S: addition and subtraction, moving from left to right.

Order of brackets: vinculum or bar (a line drawn over a group) first, then round brackets ( ), then curly brackets { }, then square brackets [ ]. We open the innermost bracket first.

Worked examples

  • 8 + 12 ÷ 4 × 2 − 3: first 12 ÷ 4 = 3, then 3 × 2 = 6, so 8 + 6 − 3 = 11.
  • 36 ÷ 6 × 3: left to right gives 6 × 3 = 18, not 36 ÷ 18 = 2.
  • 15 − [3 + {4 × (6 − 2)}]: (6 − 2) = 4; 4 × 4 = 16; 3 + 16 = 19; 15 − 19 = −4.
  • 12 + 18 ÷ 3²: the power first, 3² = 9, then 18 ÷ 9 = 2, so the answer is 14.
  • 0.2 × 0.3 + 0.4 ÷ 0.2 = 0.06 + 2 = 2.06.

Common errors: adding before multiplying; doing multiplication always before division (they are equal rank); forgetting a minus sign in front of a bracket, since −(3 + 4) = −7 but −3 + 4 = 1.

Ratio

A ratio a : b compares two quantities of the same kind in the same unit. It is the fraction a/b in the simplest form. Change units first: 15 minutes to 1 hour is 15 : 60 = 1 : 4.

Key points

  • Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio. 45 : 60 = 3 : 4.
  • a : b is not the same as b : a. 3 : 5 is different from 5 : 3.
  • Dividing a quantity N in the ratio a : b gives the shares aN/(a + b) and bN/(a + b). ₹1200 divided in 3 : 5 gives ₹450 and ₹750.
  • Combining ratios: if a : b = 2 : 3 and b : c = 4 : 5, make b equal. 2 : 3 = 8 : 12 and 4 : 5 = 12 : 15, so a : b : c = 8 : 12 : 15.
  • Compounded ratio of a : b and c : d is ac : bd. Compounded ratio of 2 : 3 and 4 : 5 is 8 : 15.
  • Duplicate ratio of a : b is a² : b². Triplicate is a³ : b³. Sub-duplicate is √a : √b.
  • Ratio and age: if the ratio of father to son is 7 : 2 and the sum is 54, then each part is 54 ÷ 9 = 6, so the son is 12.
  • Mixtures: milk and water in 3 : 2 in 40 litres gives water = 2/5 × 40 = 16 litres.
  • If two numbers are in ratio 5 : 7 and differ by 18, then 2 parts = 18, one part = 9, and the larger number is 63.

Proportion

Four numbers a, b, c, d are in proportion if a : b = c : d. a and d are the extremes; b and c are the means. The test is a × d = b × c. Check 6, 9, 10, 15: 6 × 15 = 90 and 9 × 10 = 90, so they are in proportion.

  • Fourth proportional to a, b, c is d = bc/a. For 4, 6, 10: d = 6 × 10 / 4 = 15.
  • Third proportional to a and b is b²/a. For 9 and 12: 144/9 = 16.
  • Mean proportional between a and b is √(ab). Between 4 and 25 it is 10.
  • Continued proportion: a : b = b : c = c : d.
  • Properties: if a/b = c/d then (a + b)/b = (c + d)/d (componendo) and (a − b)/b = (c − d)/d (dividendo). Also a/c = b/d (alternendo) and b/a = d/c (invertendo).

Direct and inverse proportion

  • Direct: when one quantity increases, the other increases in the same ratio; their ratio is constant. 12 pens cost ₹90, so 20 pens cost 90/12 × 20 = ₹150.
  • Inverse: when one increases the other decreases; their product is constant. 8 workers finish a job in 15 days, so 12 workers need 8 × 15 / 12 = 10 days.

Classroom angle: Use recipe, map scale and rate-of-pay stories. Let children make tables and see that in direct proportion the ratio column stays equal, and in inverse proportion the product column stays equal.

Comparison table

FeatureDirect proportionInverse proportion
What stays constantRatio y/xProduct xy
Change in x and yBoth rise or both fallOne rises, other falls
ExamplePens and costWorkers and days
Graph shapeStraight line through the originCurve (hyperbola)

Percentage

"Per cent" means "per hundred". x% = x/100. To find p% of N, compute (p/100) × N. 25% of 480 = 120.

Fraction and percentage conversion (learn these)

Fraction1/21/31/41/51/61/81/101/12
Percentage50%33⅓%25%20%16⅔%12.5%10%8⅓%

Also 3/8 = 37.5%, 12.5% = 1/8 and 75% = 3/4.

Basic types

  • What percent is A of B? (A/B) × 100. 45 is 25% of 180. 72 out of 90 is 80%.
  • Find the whole from a part: 40% of x = 60 gives x = 150.
  • Increase or decrease: new value = old × (1 ± r/100). ₹12000 raised by 15% becomes ₹13800. A number raised 10% to 66 was originally 66/1.1 = 60.
  • Percentage change = (change / original) × 100. The original is always the base.

Successive changes

  • A rise of x% followed by a rise of y% gives a net change of x + y + xy/100 per cent.
  • A rise of 20% followed by a fall of 20%: 20 − 20 − (20 × 20)/100 = −4%. There is a net 4% loss.
  • Population of 20000 rising 10% twice becomes 20000 × 1.1 × 1.1 = 24200.

Price and consumption: If the price rises by r%, the consumption must fall by r/(100 + r) × 100 per cent to keep the spending equal. For a 25% rise: 25/125 × 100 = 20%.

Comparison of A and B: If A is 25% more than B, then B is 25/125 = 20% less than A. If A is 20% less than B, then B is 25% more than A.

Marks problems: A pass mark of 40% and a student who scores 168 and fails by 12 means the pass mark is 180, so total marks = 180 / 0.4 = 450.

Useful equalities: If 30% of A = 20% of B then A : B = 20 : 30 = 2 : 3. And 25% of 25% of 400 = 25.

Exam traps

  • In 36 ÷ 6 × 3 the answer is 18, not 2, because we go left to right.
  • A ratio of 3 : 5 is not equal to 5 : 3.
  • A 20% rise and a 20% fall give a 4% net fall, not zero.
  • If A is 25% more than B, B is only 20% less than A. The percentages are not the same.
  • The mean proportional between 3 and 12 is 6 (the root of 36), not 7.5 (the average).
  • In inverse proportion you multiply the pair and keep the product, not the ratio.
  • When the base of a percentage changes (after an increase), 10% of the new value is not the same as 10% of the old value.
  • Third proportional to a and b is b²/a, not a²/b.

One-liners

  • BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
  • 45 : 60 simplifies to 3 : 4.
  • Fourth proportional to 4, 6, 10 is 15.
  • Third proportional to 9 and 12 is 16.
  • Mean proportional between 4 and 25 is 10.
  • Duplicate ratio of 3 : 4 is 9 : 16.
  • 1/6 = 16⅔%.
  • 3/8 = 37.5%.
  • 20% rise then 20% fall means a 4% net fall.
  • A 25% price rise needs a 20% cut in use to keep spending the same.
  • Compounded ratio of 2 : 3 and 4 : 5 is 8 : 15.
  • Pass mark 40%, a score 12 short of it, with a score of 168: total marks are 450.

Practice questions

  1. What is the value of 8 + 12 ÷ 4 × 2 − 3?

    1. 7
    2. 11
    3. 5
    4. 13
    Answer

    B. 11

    Division then multiplication left to right: 12 ÷ 4 = 3, 3 × 2 = 6. Then 8 + 6 − 3 = 11.

  2. What is the value of 36 ÷ 6 × 3?

    1. 2
    2. 12
    3. 18
    4. 9
    Answer

    C. 18

    Division and multiplication have equal rank, so go left to right: 6 × 3 = 18.

  3. What is the value of 15 − [3 + {4 × (6 − 2)}]?

    1. 4
    2. −4
    3. −8
    4. 8
    Answer

    B. −4

    (6 − 2) = 4; 4 × 4 = 16; 3 + 16 = 19; 15 − 19 = −4.

  4. What is the value of 12 + 18 ÷ 3²?

    1. 14
    2. 10
    3. 18
    4. 16
    Answer

    A. 14

    Power first: 3² = 9; 18 ÷ 9 = 2; 12 + 2 = 14.

  5. What is the value of 7 + 3 × (10 − 4) ÷ 2?

    1. 8
    2. 25
    3. 12
    4. 16
    Answer

    D. 16

    Bracket gives 6; 3 × 6 = 18; 18 ÷ 2 = 9; 7 + 9 = 16.

  6. What is the value of 0.2 × 0.3 + 0.4 ÷ 0.2?

    1. 2.6
    2. 0.26
    3. 2.06
    4. 0.62
    Answer

    C. 2.06

    0.2 × 0.3 = 0.06 and 0.4 ÷ 0.2 = 2, so the sum is 2.06.

  7. What is the value of (25 − 5 × 3)² ÷ 4?

    1. 25
    2. 100
    3. 50
    4. 10
    Answer

    A. 25

    5 × 3 = 15; 25 − 15 = 10; 10² = 100; 100 ÷ 4 = 25.

  8. What is 45 : 60 in its simplest form?

    1. 15 : 20
    2. 3 : 4
    3. 4 : 3
    4. 5 : 6
    Answer

    B. 3 : 4

    Divide both terms by 15 to get 3 : 4.

  9. What is the ratio of 15 minutes to 1 hour?

    1. 1 : 6
    2. 1 : 3
    3. 15 : 1
    4. 1 : 4
    Answer

    D. 1 : 4

    1 hour = 60 minutes, so 15 : 60 = 1 : 4.

  10. ₹1200 is divided in the ratio 3 : 5. What is the larger share?

    1. ₹800
    2. ₹750
    3. ₹450
    4. ₹600
    Answer

    B. ₹750

    Total parts = 8, one part = 150, so 5 parts = ₹750.

  11. What is the fourth proportional to 4, 6 and 10?

    1. 15
    2. 20
    3. 24
    4. 12
    Answer

    A. 15

    4 : 6 = 10 : x gives x = 6 × 10 / 4 = 15.

  12. What is the third proportional to 9 and 12?

    1. 18
    2. 14
    3. 16
    4. 15
    Answer

    C. 16

    9 : 12 = 12 : x gives x = 144 / 9 = 16.

  13. What is the mean proportional between 4 and 25?

    1. 20
    2. 12.5
    3. 14.5
    4. 10
    Answer

    D. 10

    √(4 × 25) = √100 = 10.

  14. If a : b = 2 : 3 and b : c = 4 : 5, then a : b : c is

    1. 2 : 3 : 5
    2. 6 : 12 : 15
    3. 8 : 12 : 20
    4. 8 : 12 : 15
    Answer

    D. 8 : 12 : 15

    Make b equal: 2 : 3 = 8 : 12 and 4 : 5 = 12 : 15.

  15. What is the duplicate ratio of 3 : 4?

    1. 27 : 64
    2. 9 : 16
    3. 3 : 8
    4. 6 : 8
    Answer

    B. 9 : 16

    Duplicate ratio of a : b is a² : b², so 9 : 16.

  16. What is the compounded ratio of 2 : 3 and 4 : 5?

    1. 8 : 15
    2. 10 : 12
    3. 6 : 20
    4. 2 : 5
    Answer

    A. 8 : 15

    Multiply the first terms and the second terms: 8 : 15.

  17. Which four numbers are in proportion?

    1. 2, 5, 8, 21
    2. 3, 5, 7, 9
    3. 6, 9, 10, 15
    4. 4, 6, 9, 12
    Answer

    C. 6, 9, 10, 15

    6 × 15 = 90 = 9 × 10, so the product of extremes equals the product of means.

  18. 8 workers can finish a job in 15 days. In how many days will 12 workers finish it, working at the same rate?

    1. 22.5 days
    2. 10 days
    3. 12 days
    4. 8 days
    Answer

    B. 10 days

    Workers and days are inversely proportional: 8 × 15 = 12 × d, so d = 10.

  19. If 12 pens cost ₹90, what is the cost of 20 pens at the same rate?

    1. ₹180
    2. ₹120
    3. ₹150
    4. ₹135
    Answer

    C. ₹150

    Cost per pen = 7.5, so 20 pens cost ₹150.

  20. Two numbers have a sum of 84 and are in the ratio 4 : 3. What is their difference?

    1. 24
    2. 6
    3. 9
    4. 12
    Answer

    D. 12

    One part = 84/7 = 12; the numbers are 48 and 36; difference = 12.

  21. Two numbers are in the ratio 5 : 7 and their difference is 18. What is the larger number?

    1. 63
    2. 54
    3. 72
    4. 45
    Answer

    A. 63

    2 parts = 18, so 1 part = 9; the larger number is 7 × 9 = 63.

  22. The ages of a father and his son are in the ratio 7 : 2 and their ages add up to 54 years. What is the son's age?

    1. 18 years
    2. 9 years
    3. 12 years
    4. 14 years
    Answer

    C. 12 years

    One part = 54 / 9 = 6, so the son is 2 × 6 = 12.

  23. A 40-litre mixture contains milk and water in the ratio 3 : 2. How much water is in it?

    1. 24 litres
    2. 20 litres
    3. 12 litres
    4. 16 litres
    Answer

    D. 16 litres

    Water = 2/5 × 40 = 16 litres.

  24. Consider these statements. 1. A ratio of two quantities of the same kind has no unit. 2. The ratio 3 : 5 is equal to the ratio 5 : 3. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    The order of terms matters, so 3 : 5 is not 5 : 3. Statement 1 is correct.

  25. Consider these statements. 1. In direct proportion the ratio of the two quantities stays constant. 2. In inverse proportion the product of the two quantities stays constant. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    These are the defining tests of direct and inverse proportion.

  26. Consider these statements. 1. The mean proportional between 3 and 12 is 9. 2. The fourth proportional to 2, 3 and 4 is 6. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    The mean proportional is √36 = 6, so 1 is wrong. 2 : 3 = 4 : x gives x = 6, so 2 is right.

  27. What is 25% of 480?

    1. 140
    2. 160
    3. 96
    4. 120
    Answer

    D. 120

    480 × 25 / 100 = 120.

  28. What is 3/8 expressed as a percentage?

    1. 37.5%
    2. 62.5%
    3. 33.3%
    4. 35%
    Answer

    A. 37.5%

    3 ÷ 8 = 0.375 = 37.5%.

  29. What percentage of 180 is 45?

    1. 40%
    2. 25%
    3. 30%
    4. 20%
    Answer

    B. 25%

    45/180 × 100 = 25.

  30. A number is increased by 20% and then decreased by 20%. What is the net change?

    1. 4% increase
    2. 4% decrease
    3. No change
    4. 2% decrease
    Answer

    B. 4% decrease

    Take 100: 120, then 20% of 120 = 24 off gives 96, a 4% fall.

  31. The price of a commodity rises by 25%. By what percentage must consumption be cut to keep spending the same?

    1. 30%
    2. 25%
    3. 15%
    4. 20%
    Answer

    D. 20%

    Fall = 25 / 125 × 100 = 20%.

  32. A salary of ₹12000 is increased by 15%. What is the new salary?

    1. ₹13800
    2. ₹14400
    3. ₹13200
    4. ₹12150
    Answer

    A. ₹13800

    15% of 12000 = 1800, so 12000 + 1800 = 13800.

  33. A student scores 168 marks and fails by 12 marks. If the pass mark is 40%, what are the maximum marks?

    1. 420
    2. 500
    3. 450
    4. 400
    Answer

    C. 450

    Pass mark = 168 + 12 = 180; 40% of T = 180, so T = 450.

  34. If 40% of a number is 60, what is the number?

    1. 24
    2. 150
    3. 240
    4. 100
    Answer

    B. 150

    x = 60 × 100 / 40 = 150.

  35. A population of 20000 grows by 10% in one year and by 10% in the next year. What is the population after two years?

    1. 24200
    2. 24000
    3. 22000
    4. 24400
    Answer

    A. 24200

    20000 → 22000 → 24200.

  36. What is 1/6 expressed as a percentage?

    1. 15%
    2. 14⅔%
    3. 16⅔%
    4. 16.5%
    Answer

    C. 16⅔%

    1 ÷ 6 = 0.1666..., which is 16⅔%.

  37. What fraction is equal to 12.5%?

    1. 1/10
    2. 1/6
    3. 1/12
    4. 1/8
    Answer

    D. 1/8

    12.5/100 = 1/8.

  38. A's income is 25% more than B's income. By what percentage is B's income less than A's?

    1. 15%
    2. 20%
    3. 30%
    4. 25%
    Answer

    B. 20%

    If B = 100, then A = 125. B is 25 less than 125, which is 20%.

  39. A student gets 72 marks out of 90. What is the percentage?

    1. 80%
    2. 72%
    3. 75%
    4. 85%
    Answer

    A. 80%

    72/90 × 100 = 80.

  40. In a class, 24 students are boys and they are 40% of the class. How many students are in the class?

    1. 48
    2. 36
    3. 60
    4. 64
    Answer

    C. 60

    40% of T = 24, so T = 24 × 100 / 40 = 60.

  41. What is the value of (20% of 50) + (50% of 20)?

    1. 10
    2. 25
    3. 30
    4. 20
    Answer

    D. 20

    20% of 50 = 10 and 50% of 20 = 10, so the sum is 20.

  42. What is 25% of 25% of 400?

    1. 50
    2. 25
    3. 12.5
    4. 100
    Answer

    B. 25

    25% of 400 = 100; 25% of 100 = 25.

  43. If 30% of A equals 20% of B, what is the ratio A : B?

    1. 2 : 3
    2. 1 : 1
    3. 3 : 2
    4. 3 : 5
    Answer

    A. 2 : 3

    0.3A = 0.2B gives A/B = 2/3.

  44. A number is increased by 10% and becomes 66. What was the original number?

    1. 55
    2. 56
    3. 60
    4. 58
    Answer

    C. 60

    Original = 66 / 1.1 = 60.

  45. Two numbers are in the ratio 3 : 4. If 6 is added to each, the ratio becomes 4 : 5. What is the smaller number?

    1. 15
    2. 12
    3. 24
    4. 18
    Answer

    D. 18

    (3k + 6)/(4k + 6) = 4/5 gives 15k + 30 = 16k + 24, so k = 6 and the smaller number is 18.

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