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State Board · Class 6 · Mathematics

Lesson 7: Ratio and Proportion

5 concepts · 40 practice questions with answers · Free to read

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What is a ratio?

From daily life

For tea, mother mixes 2 cups of water with 1 cup of milk. We compare the two amounts and say "water to milk is 2 to 1". That comparison is a ratio.

Simple definition

A ratio (गुणोत्तर) compares two quantities of the same kind. The ratio of a to b is written a : b or ab; a is the first term and b is the second term.

Rule 1: Writing a ratio — 3 pencils and 5 pens give pencils : pens = 3 : 5. If you turn it round, pens : pencils = 5 : 3, which is a different ratio.

Ratios compare parts: cement to sand is 1 : 3; peanut butter to jaggery is 5 : 2.ConcreteCcement : sand = 1 : 3Laddu mix5 peanut butter2 jaggery

In the figure : In the figure the ratio of two parts, such as 1 : 3 and 5 : 2.

  • Easy : Easy: 4 red and 7 blue balls give red : blue = 4 : 7.
  • Medium : Medium: Flour 4 cups and sugar 6 cups give 4 : 6, written as the fraction 46.
  • Exam-type : Exam: In a bag 9 are mangoes and 5 are apples. Apples : mangoes = 5 : 9, not 9 : 5.
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What is a ratio?

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Rule 2: No unit and simplest form — A ratio is a pure number and has no unit. Always write it in the simplest form: 15 : 45 = 1 : 3 (divide both by 15).

  • Easy : Easy: 10 : 20 = 1 : 2.
  • Medium : Medium: 18 : 27 = 2 : 3 (divide by 9).
  • Exam-type : Exam: Is 6 : 9 the same as 2 : 3? Dividing both by 3 gives 2 : 3, so yes.
Remember

Ratio compares two things of the same kind, has no unit, and order matters.

Common mistake

Writing the terms in the wrong order. "A to B" means A comes first.

Shortcut

Memory clue: "आधी सांगितलेले आधी लिहा".

Spot the difference

ItemMeaning
Ratio a : ba compared to b
UnitNone
Order3 : 5 and 5 : 3 differ

Practice questions

Easy
  1. 1What is a ratio?
  2. 2Write the ratio of 3 pencils to 5 pens.
  3. 3Write the ratio of pens to pencils in the same case.
Medium
  1. 4Does a ratio have a unit?
  2. 5Write 15 : 45 in the simplest form.
  3. 6Flour is 4 cups and sugar 6 cups. Find flour : sugar in the simplest form.
Challenging
  1. 7Write 12 : 20 as a fraction in the simplest form.
  2. 8Why is 2 : 3 not the same as 3 : 2?
Show answers (try them yourself first)
  1. 1A comparison of two quantities of the same kind.
  2. 23 : 5.
  3. 35 : 3.
  4. 4No, it is a pure number.
  5. 51 : 3.
  6. 64 : 6 = 2 : 3.
  7. 71220 = 35.
  8. 8The order of the terms changes which quantity is compared to which.

Simplest form and equal units

From daily life

A shopkeeper sells a pen for 3 rupees and a pencil for 60 paise. To say how much dearer the pen is, he must first change both to paise.

Simple definition

To simplify a ratio, divide both terms by their GCD (मसावि). Before comparing, make the units equal.

Rule 1: Simplest form — 24 : 36: the GCD is 12, so 24 : 36 = 2 : 3. 14 : 35: the GCD is 7, so 14 : 35 = 2 : 5.

  • Easy : Easy: 8 : 12 = 2 : 3 (GCD 4).
  • Medium : Medium: 50 : 125 = 2 : 5 (GCD 25).
  • Exam-type : Exam: 48 : 72 has GCD 24, so it is 2 : 3.

Rule 2: Equal units — 3 rupees to 60 paise: 3 rupees = 300 paise, so 300 : 60 = 5 : 1. 1.5 m to 50 cm: 1.5 m = 150 cm, so 150 : 50 = 3 : 1.

Make the units equal before finding a ratio: 5 rupees = 500 paise.5 rupees80 paise500 paise80 paise:Same unit first: 500 : 80 = 25 : 4×100

In the figure : In the figure making units equal first: rupees changed to paise.

  • Easy : Easy: 2 rupees to 50 paise = 200 : 50 = 4 : 1.
  • Medium : Medium: 2 kg to 500 g = 2000 g : 500 g = 4 : 1.
  • Exam-type : Exam: 3 hours to 45 minutes = 180 : 45 = 4 : 1.

Rule 3: Reversing the order — For 30 and 90, first : second = 30 : 90 = 1 : 3, and second : first = 90 : 30 = 3 : 1.

  • Easy : Easy: 4 and 8 give 1 : 2; the reverse is 2 : 1.
  • Medium : Medium: 20 and 60 give 1 : 3; the reverse is 3 : 1.
  • Exam-type : Exam: The ratio of 45 to 15 is 3 : 1. The ratio of 15 to 45 is 1 : 3.
Remember

Divide by the GCD. Same unit first, then compare.

Common mistake

Comparing 3 rupees with 60 paise as 3 : 60. Change to 300 : 60 first.

Shortcut

Memory clue: "आधी एकक सारखे, मग भागा".

Spot the difference

QuantitiesRatio
3 rupees, 60 paise300 : 60 = 5 : 1
1.5 m, 50 cm150 : 50 = 3 : 1
2 kg, 500 g2000 : 500 = 4 : 1

Practice questions

Easy
  1. 1Simplify 24 : 36.
  2. 2Simplify 14 : 35.
  3. 3Simplify 50 : 125.
Medium
  1. 4Find the ratio of 3 rupees to 60 paise.
  2. 5Find the ratio of 1.5 m to 50 cm.
  3. 6Find the ratio of 2 kg to 500 g.
Challenging
  1. 7Find the ratio of 3 hours to 45 minutes.
  2. 8For 30 and 90, find the second : first.
Show answers (try them yourself first)
  1. 12 : 3.
  2. 22 : 5.
  3. 32 : 5.
  4. 4300 : 60 = 5 : 1.
  5. 5150 : 50 = 3 : 1.
  6. 62000 : 500 = 4 : 1.
  7. 7180 : 45 = 4 : 1.
  8. 890 : 30 = 3 : 1.

Ratios in everyday life

From daily life

In a cricket team photo, 12 of the 28 students are boys. We can say boys to girls, or girls to all students. Each question gives a different ratio.

Simple definition

Ratios can compare a part with a part, or a part with the whole. If a part is missing, find it first.

Rule 1: A shaded grid — A grid has 15 cells and 6 are shaded. Unshaded cells = 15 − 6 = 9. Shaded : unshaded = 6 : 9 = 2 : 3 and shaded : total = 6 : 15 = 2 : 5.

A 20-cell grid with 8 shaded: shaded to unshaded is 8 : 12 = 2 : 3.Total 20 cellsShaded 8, unshaded 12Shaded : unshaded = 2 : 3Shaded : total = 2 : 5

In the figure : In the figure a grid with some cells shaded and the ratios 2 : 3 and 2 : 5.

  • Easy : Easy: 10 cells, 4 shaded: shaded : unshaded = 4 : 6 = 2 : 3.
  • Medium : Medium: 20 cells, 5 shaded: shaded : total = 5 : 20 = 1 : 4.
  • Exam-type : Exam: 25 cells, 15 unshaded: shaded = 10, so shaded : unshaded = 10 : 15 = 2 : 3.

Rule 2: A class and a family budget — A class of 40 has 16 cricket, 14 football and 10 kho-kho players. Cricket : kho-kho = 16 : 10 = 8 : 5 and football : total = 14 : 40 = 7 : 20. A family has income ₹ 45,000 and savings ₹ 9,000, so expenditure = ₹ 36,000 and income : expenditure = 45 : 36 = 5 : 4.

  • Easy : Easy: 12 boys among 28 students: girls = 16, boys : girls = 12 : 16 = 3 : 4.
  • Medium : Medium: Savings : income = 9 : 45 = 1 : 5.
  • Exam-type : Exam: Girls : total = 16 : 28 = 4 : 7.
Remember

Part to part or part to whole. Find the missing part first.

Common mistake

Using the total instead of the other part. Read the question: "to unshaded" or "to total"?

Shortcut

Memory clue: "भाग की पूर्ण? आधी ठरवा".

Spot the difference

ComparisonExample
Part : partshaded : unshaded = 2 : 3
Part : wholeshaded : total = 2 : 5
Missing partFind it first

Practice questions

Easy
  1. 1A grid has 15 cells and 6 are shaded. How many are unshaded?
  2. 2Find shaded : total for the same grid.
  3. 3Of 28 students, 12 are boys. Find boys : girls.
Medium
  1. 4In a class of 40, 16 play cricket and 10 play kho-kho. Find cricket : kho-kho.
  2. 5In the same class 14 play football. Find football : total.
  3. 6Income is ₹ 45,000 and savings ₹ 9,000. Find savings : income.
Challenging
  1. 7For the same family find income : expenditure.
  2. 8Of 28 students, 12 are boys. Find girls : total.
Show answers (try them yourself first)
  1. 19.
  2. 26 : 15 = 2 : 5.
  3. 312 : 16 = 3 : 4.
  4. 416 : 10 = 8 : 5.
  5. 514 : 40 = 7 : 20.
  6. 69 : 45 = 1 : 5.
  7. 7Expenditure = 45,000 − 9,000 = 36,000; 45 : 36 = 5 : 4.
  8. 816 : 28 = 4 : 7.

Proportion

From daily life

Two photos look the same shape if one is just a smaller copy of the other. A frame 80 mm by 60 mm and a frame 40 mm by 30 mm have the same ratio, so the pictures look alike.

Simple definition

Two equal ratios are said to be in proportion (प्रमाणात).

Rule 1: Testing proportion — Reduce both ratios to the simplest form and compare. 6 : 8 = 3 : 4 and 9 : 12 = 3 : 4, so they are in proportion. 3 : 4 and 6 : 9 (= 2 : 3) are not equal.

Frames A and C have equal ratios, 60 : 40 = 30 : 20 = 3 : 2.A60 : 40B60 : 20C30 : 20D30 : 40A and C: both 3 : 2, so they look the same

In the figure : In the figure frames in proportion, with the same ratio 3 : 2.

  • Easy : Easy: 2 : 4 and 3 : 6 are both 1 : 2, so in proportion.
  • Medium : Medium: 4 : 6 and 10 : 15 are both 2 : 3, so in proportion.
  • Exam-type : Exam: Frames 80 mm by 60 mm and 40 mm by 25 mm: 4 : 3 and 8 : 5 differ, so not in proportion.

Rule 2: Finding a missing term — Multiply or divide both terms by the same number. 4 : 9 = 12 : ? Here 4 was multiplied by 3, so 9 × 3 = 27. And 7 : 2 = 35 : ? gives 2 × 5 = 10.

  • Easy : Easy: 1 : 3 = 2 : ? gives 6.
  • Medium : Medium: 5 : 8 = ? : 24 gives 15 (8 × 3 = 24, so 5 × 3).
  • Exam-type : Exam: 7 : 2 = 35 : 10 because both were multiplied by 5.
Remember

Equal ratios are in proportion. Scale both terms by the same number.

Common mistake

Multiplying only one term. Both terms change by the same number.

Shortcut

Memory clue: "दोन्हीला सारखेच गुणा".

Spot the difference

PairIn proportion?
6 : 8 and 9 : 12Yes (3 : 4)
3 : 4 and 6 : 9No
4 : 6 and 10 : 15Yes (2 : 3)

Practice questions

Easy
  1. 1Are 6 : 8 and 9 : 12 in proportion?
  2. 2Are 4 : 6 and 10 : 15 in proportion?
  3. 3Are 3 : 4 and 6 : 9 in proportion?
Medium
  1. 4Find ? in 4 : 9 = 12 : ?.
  2. 5Find ? in 5 : 8 = ? : 24.
  3. 6Find ? in 7 : 2 = 35 : ?.
Challenging
  1. 7Are frames 80 mm by 60 mm and 40 mm by 30 mm in proportion?
  2. 8Are frames 80 mm by 60 mm and 40 mm by 25 mm in proportion?
Show answers (try them yourself first)
  1. 1Yes, both are 3 : 4.
  2. 2Yes, both are 2 : 3.
  3. 3No, 3 : 4 differs from 2 : 3.
  4. 427.
  5. 515.
  6. 610.
  7. 7Yes, both are 4 : 3.
  8. 8No, 4 : 3 differs from 8 : 5.

The unitary method

From daily life

If 12 bananas cost ₹ 60, how much will 5 bananas cost? First find the cost of one banana, then multiply. This is the unitary method.

Simple definition

The unitary method (ऐकिक पद्धत) first finds the value of one unit by division, then the value of many units by multiplication.

Rule 1: Cost problems — 15 pens cost ₹ 90, so 1 pen costs 90 ÷ 15 = ₹ 6 and 8 pens cost 8 × 6 = ₹ 48. 12 bananas for ₹ 60 give ₹ 5 each, so 5 bananas cost ₹ 25.

Unitary method: divide to get the price of one, multiply to get the price of many.12 saplings₹ 1441 sapling₹ 1210 saplings₹ 120÷ 12× 10Unitary method: find one, then find many

In the figure : In the figure the unitary method: divide first, then multiply.

  • Easy : Easy: 6 notebooks cost ₹ 72, so 1 costs ₹ 12.
  • Medium : Medium: 9 such notebooks cost 9 × 12 = ₹ 108.
  • Exam-type : Exam: 12 eggs cost ₹ 96, so 30 eggs cost 30 × 8 = ₹ 240.

Rule 2: Speed and rice — A car goes 180 km in 3 hours, so 60 km in 1 hour and 5 hours give 5 × 60 = 300 km. 20 kg of rice costs ₹ 800, so 1 kg costs ₹ 40 and 7 kg cost ₹ 280.

  • Easy : Easy: 180 km in 3 hours is 60 km in 1 hour.
  • Medium : Medium: At that speed, 5 hours give 300 km.
  • Exam-type : Exam: 20 kg of rice for ₹ 800 means ₹ 40 per kg, so 7 kg cost ₹ 280.
Remember

Divide to get one, multiply to get many.

Common mistake

Multiplying first and dividing later by mistake. First find the value of one.

Shortcut

Memory clue: "आधी एक, मग अनेक".

Spot the difference

StepAction
1Divide to find the value of 1
2Multiply for the number asked

Practice questions

Easy
  1. 16 notebooks cost ₹ 72. Find the cost of 1.
  2. 2Find the cost of 9 such notebooks.
  3. 315 pens cost ₹ 90. Find the cost of 8 pens.
Medium
  1. 4A car goes 180 km in 3 hours. How far in 1 hour?
  2. 5How far will it go in 5 hours?
  3. 620 kg of rice costs ₹ 800. Find the cost of 7 kg.
Challenging
  1. 712 eggs cost ₹ 96. Find the cost of 30 eggs.
  2. 8Why do we find the value of one unit first?
Show answers (try them yourself first)
  1. 1₹ 12.
  2. 29 × 12 = ₹ 108.
  3. 390 ÷ 15 = 6; 8 × 6 = ₹ 48.
  4. 460 km.
  5. 55 × 60 = 300 km.
  6. 6800 ÷ 20 = 40; 7 × 40 = ₹ 280.
  7. 796 ÷ 12 = 8; 30 × 8 = ₹ 240.
  8. 8Once we know one, any number of units is found by multiplying.