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State Board · Class 9 · Mathematics Part 1

Lesson 4: Ratio and Proportion

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

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Ratio and its properties

From daily life

In chai, 2 cups of milk to 1 cup of water gives the same taste whether you make 3 cups or 30 cups, as long as the 2 : 1 is kept. That steady comparison is a ratio.

Simple definition

A ratio (गुणोत्तर) compares two quantities of the same kind by division: a : b or ab.

Rule 1: Parts of a ratio — In a : b, a is the predecessor (पूर्वपद, first term) and b the successor (उत्तरपद, second term). If the second term is 100, the ratio is a percentage: 3 : 4 = 75 : 100 = 75%.

  • Easy : Easy: 4 : 6 = 2 : 3.
  • Medium : Medium: 9 : 12 = 3 : 4, which is 75%.
  • Exam-type : Exam: Express 18 marks out of 24 as a percentage: 18 : 24 = 3 : 4 = 75%.

Rule 2: Equal ratios and units — A ratio does not change if both terms are multiplied or divided by the same non-zero number: a : b = ak : bk, so 2 : 5 = 4 : 10 = 6 : 15. The quantities must be in the same unit, and the ratio itself has no unit: 1.5 km and 500 m give 1500 : 500 = 3 : 1.

Two bars with the same proportion of juice and water: 3 to 5 and 6 to 10Mixing juice and water: 3 : 5Juice 3 parts, water 5 partsDouble the mix: 6 parts : 10 parts3 : 5 = 6 : 10 – same taste, same ratio

In the figure : In the figure a ratio does not change when both parts are multiplied by the same number.

  • Easy : Easy: 3 : 7 = 6 : 14.
  • Medium : Medium: 2 hours and 40 minutes give 120 : 40 = 3 : 1.
  • Exam-type : Exam: Ratio of 2.5 kg to 750 g: 2500 : 750 = 10 : 3.

Rule 3: Three or more quantities — Numbers in the ratio 3 : 5 : 2 with sum 50: 3m + 5m + 2m = 10m = 50, so m = 5 and the numbers are 15, 25, 10. If A : B = 2 : 3 and B : C = 4 : 5, make B equal: A : B : C = 8 : 12 : 15.

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Ratio and its properties

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  • Easy : Easy: 1 : 2 : 3 with sum 12 gives 2, 4, 6.
  • Medium : Medium: A : B = 3 : 4, B : C = 4 : 7 gives A : B : C = 3 : 4 : 7.
  • Exam-type : Exam: Red : blue marbles = 3 : 4. After adding 6 of each it becomes 5 : 6. Then (3x + 6) : (4x + 6) = 5 : 6, so 18x + 36 = 20x + 30, x = 3. Marbles: 9 and 12.
Remember

A ratio compares same units and has no unit. Multiply or divide both terms alike.

Common mistake

Comparing 2 hours with 40 minutes as 2 : 40. Convert first: 120 : 40.

Shortcut

Memory clue: "आधी एकक एक, मग गुणोत्तर".

Spot the difference

RatioEqual ratioNote
2 : 54 : 10Multiply by 2
18 : 243 : 4Divide by 6
1500 : 5003 : 1Same unit first

Practice questions

Easy
  1. 1What is a ratio?
  2. 2Name the two terms of a : b.
  3. 3Simplify 15 : 20.
Medium
  1. 4Write 3 : 5 as a percentage.
  2. 5Ratio of 3 m to 60 cm?
  3. 6Divide 60 in the ratio 1 : 2 : 3.
Challenging
  1. 7A : B = 2 : 3 and B : C = 5 : 4. Find A : B : C.
  2. 8Two numbers are in the ratio 2 : 3. Adding 4 to each gives 3 : 4. Find them.
Show answers (try them yourself first)
  1. 1A comparison of two like quantities by division.
  2. 2Predecessor and successor.
  3. 33 : 4.
  4. 460 : 100 = 60%.
  5. 5300 : 60 = 5 : 1.
  6. 610, 20, 30.
  7. 710 : 15 : 12.
  8. 8(2x + 4) : (3x + 4) = 3 : 4 gives 8x + 16 = 9x + 12, x = 4; numbers 8 and 12.

Direct and inverse proportion, comparison of ratios

From daily life

If 1 notebook costs ₹ 10, then 3 notebooks cost ₹ 30. More notebooks, more money. But if more workers join a job, the days needed fall. These are two different kinds of change.

Simple definition

In direct proportion (प्रत्यक्ष चल) the ratio of two quantities stays constant. In inverse proportion (व्यस्त चल) their product stays constant.

Rule 1: Direct proportion — If x and y are in direct proportion, xy = k, a constant. Both rise or fall together. 5 pencils cost ₹ 30 and 8 pencils cost ₹ 48: 305 = 488 = 6 per pencil.

Left: a straight-line graph of cost against number of pens. Right: a falling curve of days against number of workersDirect and inverse proportionPens (number)CostDirect: more pens, more coststraight line through 0Workers (number)DaysInverse: more workers, fewer dayscurve falls

In the figure : In the figure in direct proportion both quantities rise together; in inverse proportion one rises as the other falls.

  • Easy : Easy: 2 notebooks cost ₹ 20, so 6 notebooks cost ₹ 60.
  • Medium : Medium: 12 km in 3 hours means 4 km per hour, so 5 hours give 20 km.
  • Exam-type : Exam: A tap fills 45 litres in 9 minutes. In 14 minutes: 45 × 149 = 70 litres.

Rule 2: Inverse proportion — If x and y are in inverse proportion, x × y = k. One rises as the other falls. 6 workers finish a job in 10 days, so 15 workers need 6015 = 4 days (product 60).

  • Easy : Easy: 4 taps fill a tank in 9 hours, so 6 taps need 36 ÷ 6 = 6 hours.
  • Medium : Medium: At 40 km/h a trip takes 3 hours, so at 60 km/h it takes 120 ÷ 60 = 2 hours.
  • Exam-type : Exam: 8 men build a wall in 15 days. 12 men need 8 × 1512 = 10 days.

Rule 3: Comparing ratios — For positive a, b, c, d: if ad > bc then ab > cd; if ad < bc then ab < cd; if ad = bc they are equal. 35 and 58: 3 × 8 = 24 and 5 × 5 = 25, so 35 < 58. 79 and 57: 49 > 45, so 79 > 57.

  • Easy : Easy: 12 > 13 since 1 × 3 = 3 > 2 = 1 × 2.
  • Medium : Medium: 47 < 35 since 20 < 21.
  • Exam-type : Exam: Two numbers are in the ratio 3 : 4. Adding 10 to each gives 5 : 6. (3x + 10) : (4x + 10) = 5 : 6 gives 18x + 60 = 20x + 50, x = 5; numbers 15 and 20.
Remember

Direct: ratio is constant. Inverse: product is constant.

Common mistake

Using the ratio method for workers and days. More workers means fewer days, so use the product.

Shortcut

Memory clue: "दोन्ही वाढले = भागाकार; एक वाढले एक घटले = गुणाकार".

Spot the difference

TypeConstantExample
Directx ÷ yCost and quantity
Inversex × yWorkers and days
Inversex × ySpeed and time

Practice questions

Easy
  1. 1In direct proportion, what stays constant?
  2. 2In inverse proportion, what stays constant?
  3. 33 pens cost ₹ 24. Cost of 5 pens?
Medium
  1. 45 men finish a job in 12 days. Days for 10 men?
  2. 5Which is larger: 23 or 35?
  3. 6A car uses 6 litres for 90 km. Petrol for 150 km?
Challenging
  1. 7Is speed against time (fixed distance) direct or inverse?
  2. 8Two numbers are in the ratio 2 : 5. Adding 6 to each gives 1 : 2. Find them.
Show answers (try them yourself first)
  1. 1The ratio.
  2. 2The product.
  3. 3₹ 40.
  4. 46 days.
  5. 523, since 10 > 9.
  6. 610 litres.
  7. 7Inverse.
  8. 82(2x + 6) = 5x + 6 gives x = 6; numbers 12 and 30.

Operations on equal ratios

From daily life

If a recipe is doubled, every ingredient doubles but the taste stays equal. Rules like invertendo and alternando let us flip and swap equal ratios without breaking the equality.

Simple definition

If ab = cd, the rules invertendo, alternando, componendo, dividendo and componendo-dividendo give new equal ratios.

Rule 1: The five rules — Invertendo: ba = dc. Alternando: ac = bd. Componendo: a+bb = c+dd. Dividendo: a−bb = c−dd. Componendo-dividendo: a+ba−b = c+dc−d.

Five operations on equal ratios shown with 2/3 = 4/6If 2/3 = 4/6 then…Invertendo3/2 = 6/4Alternando2/4 = 3/6Componendo(2+3)/3 = (4+6)/6 → 5/3 = 10/6Dividendo(2−3)/3 = (4−6)/6 → −1/3 = −1/3Componendo-dividendo(2+3)/(2−3) = (4+6)/(4−6) → −5 = −5

In the figure : In the figure try each rule on 2/3 = 4/6 and see that both sides stay equal.

  • Easy : Easy: 23 = 46 gives 32 = 64 (invertendo).
  • Medium : Medium: 23 = 46 gives 24 = 36 (alternando).
  • Exam-type : Exam: 23 = 46 gives 53 = 106 (componendo) and −13 = −26 (dividendo).

Rule 2: Using the rules — If ab = 32: a+bb = 52; a−bb = 12; a+ba−b = 5. If ab = 52, put a = 5, b = 2: 3a+2b3a−2b = 1911. If xy = 35: 5x−2y5x+2y = 525 = 15.

  • Easy : Easy: ab = 21 gives a+bb = 3.
  • Medium : Medium: ab = 43 gives a+bb = 73.
  • Exam-type : Exam: ab = 34 gives 2a+b2a−b = 102 = 5.

Rule 3: Solving with componendo-dividendo — Solve x+3x−3 = 75. Apply componendo-dividendo: 2x6 = 122 = 6, so x = 18. Check: 2115 = 75.

  • Easy : Easy: x+1x−1 = 31 gives 2x2 = 42, so x = 2.
  • Medium : Medium: x+2x−2 = 53 gives 2x4 = 82 = 4, so x = 8.
  • Exam-type : Exam: x+5x−5 = 32 gives 2x10 = 51, so x = 25.
Remember

Componendo adds the bottom to the top. Dividendo subtracts it.

Common mistake

Using these rules on ratios that are not equal. Always start from ab = cd.

Shortcut

Memory clue: "योग = जोड, वियोग = वजा, दोन्ही = योग-वियोग".

Spot the difference

RuleResult
Invertendoba = dc
Alternandoac = bd
Componendoa+bb = c+dd
Dividendoa−bb = c−dd

Practice questions

Easy
  1. 1Starting condition for these rules?
  2. 2Invertendo of 35 = 610?
  3. 3Alternando of 35 = 610?
Medium
  1. 4Componendo of ab = cd?
  2. 5If ab = 51, find a+bb.
  3. 6If ab = 21, find a+ba−b.
Challenging
  1. 7Solve x+4x−4 = 53.
  2. 8Why does componendo work?
Show answers (try them yourself first)
  1. 1ab = cd.
  2. 253 = 106.
  3. 336 = 510.
  4. 4a+bb = c+dd.
  5. 56.
  6. 63.
  7. 72x8 = 82 = 4, so x = 16.
  8. 8Add 1 to both sides of ab = cd.

Theorem of equal ratios

From daily life

Three classrooms each have boys : girls = 4 : 3. Together the whole school also has 4 : 3. Adding the tops and adding the bottoms keeps the same ratio.

Simple definition

Theorem of equal ratios (समान गुणोत्तरांचे प्रमेय): if ab = cd then each ratio equals a+cb+d.

Rule 1: The theorem and its proof — If ab = cd then ab = cd = a+cb+d. Proof: let ab = cd = k. Then a = bk and c = dk, so a+cb+d = bk+dkb+d = k.

The ratios 2/3, 4/6 and 6/9 are equal, and adding all tops and all bottoms gives 12/18 which is again 2/3Theorem of equal ratios2/3=4/6=6/9=2/3add all tops and all bottoms:(2 + 4 + 6) / (3 + 6 + 9) = 12/18 = 2/3 ✓

In the figure : In the figure equal ratios stay the same when all their tops and all their bottoms are added.

  • Easy : Easy: 23 = 46 = 69, and 69 = 23.
  • Medium : Medium: 35 = 915 gives 1220 = 35.
  • Exam-type : Exam: If x2 = y3 = z5 and x + y + z = 40, each ratio is 4010 = 4, so x = 8, y = 12, z = 20.

Rule 2: General form — If ab = cd = ef, each ratio also equals la+mc+nelb+md+nf for any non-zero l, m, n. Example: a2 = b3 gives each ratio = 5a+4b22. And x3 = y5 = z7 gives each = 2x+3y−z14.

  • Easy : Easy: Classes with 4 : 3, 8 : 6 and 12 : 9 give 24 : 18 = 4 : 3 together.
  • Medium : Medium: a2 = b3 gives 5a+4b22 = k; check with a = 2k, b = 3k: 10k + 12k = 22k.
  • Exam-type : Exam: x3 = y5 = z7 = k gives 2x + 3y − z = 6k + 15k − 7k = 14k, so the ratio is 2x+3y−z14.
Remember

Add all tops and all bottoms. Same ratio. Use k to prove.

Common mistake

Adding only some terms, or adding tops and subtracting bottoms. Do the same on top and bottom.

Shortcut

Memory clue: "अंशांची बेरीज ÷ छेदांची बेरीज = तेच गुणोत्तर".

Spot the difference

GivenResult
ab = cda+cb+d = same
x2 = y3 = z5x+y+z10 = same
a2 = b35a+4b22 = same

Practice questions

Easy
  1. 1State the theorem of equal ratios.
  2. 2What do we let each ratio equal in the proof?
  3. 312 = 36. Find 1+32+6.
Medium
  1. 4If x4 = y6 and x + y = 20, find x.
  2. 5If x2 = y3 = z5 and x + y + z = 40, find z.
  3. 6Two schools have boys : girls = 5 : 4 and 10 : 8. What is it together?
Challenging
  1. 7If a3 = b4, show each ratio is 2a+5b26.
  2. 8Why is the k-method useful in proofs?
Show answers (try them yourself first)
  1. 1ab = cd = a+cb+d.
  2. 2k.
  3. 348 = 12.
  4. 4Ratio = 2, so x = 8.
  5. 520.
  6. 615 : 12 = 5 : 4.
  7. 7a = 3k, b = 4k: 6k + 20k = 26k.
  8. 8It turns every equal ratio into a = bk, so terms can be replaced.

Continued proportion and the k-method

From daily life

In 3, 6, 12 each number is double the one before. The middle number 6 is the "balance point" between 3 and 12 because 6 × 6 = 3 × 12.

Simple definition

If ab = bc, then a, b, c are in continued proportion (संतत प्रमाण), and b is the geometric mean (भूमितीय मध्य) of a and c.

Rule 1: Continued proportion — a, b, c are in continued proportion when ab = bc, that is ac = b². Then b = √(ac). 2, 6, 18: 6² = 36 = 2 × 18. The geometric mean of 9 and 16 is √144 = 12.

The numbers 2, 6, 18 each multiply by 3; the square of the middle number equals the product of the first and lastContinued proportion2618× 3× 32 : 6 = 6 : 18 (each ratio is 1 : 3)Middle number² = first × last: 6² = 2 × 18 = 366 is the geometric mean of 2 and 18

In the figure : In the figure in a continued proportion the middle number is the geometric mean of the other two.

  • Easy : Easy: 1, 3, 9 are in continued proportion since 3² = 9 = 1 × 9.
  • Medium : Medium: The geometric mean of 4 and 25 is √100 = 10.
  • Exam-type : Exam: Find x if 3, x, 27 are in continued proportion: x² = 81, so x = 9.

Rule 2: Finding the third term — If 2a, 6a², q are in continued proportion, (6a²)² = 2a × q, so 36a⁴ = 2aq and q = 18a³.

  • Easy : Easy: 2, 4, p gives 16 = 2p, so p = 8.
  • Medium : Medium: 5, 15, p gives 225 = 5p, so p = 45.
  • Exam-type : Exam: 3a, 6a², q give 36a⁴ = 3a × q, so q = 12a³.

Rule 3: The k-method — Put each equal ratio equal to k. If a, b, c are in continued proportion, b = ck and a = bk = ck². Four terms in continued proportion are a, ak, ak², ak³. With first term 3 and last term 81, 3k³ = 81, k³ = 27, k = 3, so the terms are 3, 9, 27, 81.

  • Easy : Easy: With first term 2 and k = 3 the terms are 2, 6, 18.
  • Medium : Medium: First term 5 and k = 2 give 5, 10, 20, 40.
  • Exam-type : Exam: If a, b, c are in continued proportion, a+bb+c = ck²+ckck+c = ck(k+1)c(k+1) = k = ab.
Remember

Continued proportion: ac = b². Use a = ck², b = ck.

Common mistake

Taking b = (a + c) ÷ 2. That is the arithmetic mean. The geometric mean is √(ac).

Shortcut

Memory clue: "मधला वर्ग = बाहेरच्यांचा गुणाकार".

Spot the difference

TermFormulaExample
Continued proportionac = b²2, 6, 18
Geometric meanb = √(ac)4 and 25 give 10
k-methoda = ck², b = ck3, 9, 27, 81

Practice questions

Easy
  1. 1When are a, b, c in continued proportion?
  2. 2What is the middle term called?
  3. 3Are 2, 4, 8 in continued proportion?
Medium
  1. 4Geometric mean of 2 and 8?
  2. 5Find x if 4, x, 36 are in continued proportion.
  3. 6Four terms in continued proportion start 2 with k = 3. List them.
Challenging
  1. 7If a, b, c are in continued proportion, write a and b in terms of c and k.
  2. 8Is 6 the arithmetic or geometric mean of 3 and 12?
Show answers (try them yourself first)
  1. 1When ab = bc, i.e. ac = b².
  2. 2Geometric mean (mean proportional).
  3. 3Yes, 4² = 16 = 2 × 8.
  4. 44.
  5. 5x² = 144, so x = 12.
  6. 62, 6, 18, 54.
  7. 7b = ck, a = ck².
  8. 8Geometric (6² = 36 = 3 × 12); the arithmetic mean is 7.5.

Quick revision

  • a : b = ak : bk; direct proportion has constant ratio, inverse proportion has constant product.
  • Invertendo, alternando, componendo, dividendo and componendo-dividendo; ab = cd = a+cb+d.
  • Continued proportion: ac = b²; b is the geometric mean of a and c.