MathsCBSE Class 8 · Proportional Reasoning-1

Sharing in a Ratio, Mixtures and Unit Conversions for Class 8

Updated 17 October 2026 · Fact-checked

To divide a quantity x in the ratio m : n, split it into m + n equal groups of size x/(m + n); the shares are m groups and n groups. The same method handles three-part ratios, profit sharing by investment and mixtures; when one part of a mixture is added, the other part stays fixed. Unit conversions use a given factor, such as 1 hectare = 10,000 square metres and 1 acre = 43,560 square feet.

Add the ratio parts (4+3=7) to get equal groups, find one group (42÷7=6), then multiply by each part to get the shares.

Illustration drawn for QuestionBuddy.

Understand Sharing in a Ratio, Mixtures and Unit Conversions

Sharing in a ratio. To share 12 counters equally, the ratio is $1 : 1$ . To share them in the ratio $3 : 1$ you can deal 3 and 1 repeatedly: 9 and 3. For large quantities, use groups.

*Method for dividing $x$ in the ratio $m : n$ :*
1. Total number of equal groups $= m + n$ .
2. Size of each group $= \frac{x}{m + n}$ .
3. First share $= m \times \frac{x}{m + n}$ , second share $= n \times \frac{x}{m + n}$ .
For 42 counters in the ratio $4 : 3$ : groups $= 7$ , group size $= 6$ , shares $24$ and $18$ . The method extends to three or more parts: for $a : b : c$ there are $a + b + c$ groups.

Profit sharing. Partners share profit in the ratio of their investments (when the investments last equally long). Simplify the ratio of investments first. If Prashanti invests 75,000 and Bhuvan 25,000, the ratio is $3 : 1$ , so a profit of 4,000 splits as 3,000 and 1,000.

Mixtures. A mixture of sand and cement in the ratio $3 : 1$ weighing 40 kg has $\frac{3}{4} \times 40 = 30$ kg of sand and $10$ kg of cement. If you add one ingredient, the amount of the other stays the same, so the new ratio is found by fixing the unchanged part. To change the ratio to $5 : 2$ with 30 kg of sand fixed, the cement must be $12$ kg, so 2 kg is added.

Unit conversions. Proportional problems often need units changed. The chapter gives these conversion facts:

1 metre = 3.281 feet.
1 square metre = 10.764 square feet.
1 acre = 43,560 square feet.
1 hectare = 10,000 square metres.
1 hectare = 2.471 acres.
1 millilitre = 1 cubic centimetre; 1 litre = 1,000 mL = 1,000 cubic centimetres.

Multiply by the conversion factor to go to the smaller unit; divide to go to the bigger unit. Because these factors are rounded, conversions by different routes agree only approximately. Temperature is different: 0 degrees Celsius is 32 degrees Fahrenheit, and the conversion is not a pure multiplication.

Ratios, simplest form and unit rates are on the previous topic page of this chapter.

Sharing in a Ratio, Mixtures and Unit Conversions: formulas to remember

Dividing in a ratio m : n: $shares = \frac{m}{m + n} x, \frac{n}{m + n} x$; Each group has size x/(m + n).
Three-part ratio: $a : b : c \Rightarrow \frac{a}{a + b + c} x, \frac{b}{a + b + c} x, \frac{c}{a + b + c} x$; Add the three terms to get the number of groups.
Profit sharing: $profit ratio = investment ratio$; When the time invested is the same.
Hectare and square metres: $1 hectare = 10, 000 m^{2}$; Also 1 hectare = 2.471 acres.
Acre and square feet: $1 acre = 43, 560 sq ft$; And 1 sq m = 10.764 sq ft.

How to solve Sharing in a Ratio, Mixtures and Unit Conversions questions

Follow these steps for sharing, mixture and conversion problems.

1Simplify the given ratio (for example investments 75,000 : 25,000 to 3 : 1).
2Add the terms to get the number of equal groups and divide the total by it to find the size of one group.
3Multiply the group size by each term to get each share; check that the shares add up to the total.
4In a mixture, find each ingredient first. When one ingredient is added, keep the other constant and use the new ratio to find the changed amount.
5For conversions, write the factor, choose multiply or divide by checking whether the answer should be larger or smaller, and keep a sensible number of digits.
6State units in the answer.

Sharing in a Ratio, Mixtures and Unit Conversions in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Dividing a quantity in a ratio

Suppose a family shares a 840 rupee grocery bill between two households in the ratio 3 : 4 according to the number of members.

How much does each household pay?

Try it first, then show the solutionSolution

Groups $= 3 + 4 = 7$ ; each group $= 840 \div 7 = 120$ rupees.
First pays $3 \times 120 = 360$ rupees; second pays $4 \times 120 = 480$ rupees.

Answer: 360 rupees and 480 rupees

Memory hook: Total divided by the sum of the ratio terms gives one group.

Converting land area

Imagine a plot of 2.5 acres that must be described in hectares for a government form.

Convert 2.5 acres to square feet and to hectares.

Try it first, then show the solutionSolution

Square feet $= 2.5 \times 43, 560 = 1, 08, 900$ .
Hectares $= 2.5 \div 2.471 \approx 1.01$ .

Answer: 108,900 square feet; about 1.01 hectares

Memory hook: To go from acres to hectares, divide by 2.471 because a hectare is the bigger unit.

Tricks to remember Sharing in a Ratio, Mixtures and Unit Conversions

Short memory aids. Say each one aloud once, then use it on a question the same day.

Add the terms for groups
For m : n the whole is m + n groups. Divide, then multiply.
Use it when: Every sharing problem.
Added part changes, the other stays
When you add only one ingredient, the other ingredient's amount is constant; build the equation on it.
Use it when: Mixture problems.
Bigger unit, smaller number
Hectares are bigger than square metres, so the same land has fewer hectares than square metres.
Use it when: Deciding to multiply or divide in conversions.
Shares add up
After sharing, add the shares and compare with the total.
Use it when: Checking answers.

Sharing in a Ratio, Mixtures and Unit Conversions: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Larger share · 1 mark · MCQBoard level

Rs 840 is divided in the ratio 3 : 4. The larger share is: (a) Rs 360 (b) Rs 420 (c) Rs 480 (d) Rs 560

Try it first, then show the solutionSolution

Groups $= 7$ , each $= 840 \div 7 = 120$ .
Larger share $= 4 \times 120 = 480$ .

Answer: (c) Rs 480

Example 2: Parts of a mixture · 2 marks · Short answerBoard level

A 56 L mixture of milk and water is in the ratio 5 : 3. Find the amount of each.

Try it first, then show the solutionSolution

Groups $= 8$ ; each $= 56 \div 8 = 7$ L. (1 mark)
Milk $= 5 \times 7 = 35$ L and water $= 3 \times 7 = 21$ L. (1 mark)

Answer: Milk 35 L, water 21 L

Example 3: Profit sharing with a new partner · 3 marks · Short answerHarder board level

Anita invests 45,000 and Binod 30,000 and earn a profit of 7,500. Find each share. If Chitra had joined with 15,000 from the start and the profit were 9,000, find the three shares.

Try it first, then show the solutionSolution

$45000 : 30000 = 3 : 2$ ; groups $= 5$ ; each $= 7500 \div 5 = 1500$ . Shares: $4500$ and $3000$ . (1 mark)
With Chitra: $45 : 30 : 15 = 3 : 2 : 1$ . (1 mark)
Groups $= 6$ , each $= 9000 \div 6 = 1500$ ; shares $4500$ , $3000$ , $1500$ . (1 mark)

Answer: First case: 4,500 and 3,000; with Chitra: 4,500, 3,000 and 1,500.

Example 4: Changing a mixture by adding one part · 3 marks · Short answerHarder board level

A 50 kg mixture contains rice flour and wheat flour in the ratio 2 : 3. How much rice flour must be added to make the ratio 1 : 1? To make it 3 : 2?

Try it first, then show the solutionSolution

Rice flour $= \frac{2}{5} \times 50 = 20$ kg, wheat flour $= 30$ kg. The wheat flour stays 30 kg. (1 mark)
For $1 : 1$ the rice flour must be 30 kg, so add $30 - 20 = 10$ kg. (1 mark)
For $3 : 2$ : rice flour $= 30 \times \frac{3}{2} = 45$ kg, so add $45 - 20 = 25$ kg. (1 mark)

Answer: Add 10 kg for 1 : 1 and 25 kg for 3 : 2.

Example 5: Area conversions · 4 marks · Long answerHarder board level

A plot is 2.5 acres. Find its area in square feet, in hectares (to two decimal places) and in square metres (to the nearest hundred), using 1 acre = 43,560 sq ft, 1 hectare = 2.471 acres, 1 hectare = 10,000 sq m.

Try it first, then show the solutionSolution

Square feet $= 2.5 \times 43560 = 108900$ . (1 mark)
Hectares $= 2.5 \div 2.471 \approx 1.0117 \approx 1.01$ . (1 mark)
Square metres $= 1.0117 \times 10000 \approx 10117$ . (1 mark)
To the nearest hundred, about 10,100 sq m. (Check with 1 sq m = 10.764 sq ft: $108900 \div 10.764 \approx 10117$ .) (1 mark)

Answer: 108,900 sq ft; about 1.01 hectares; about 10,100 sq m

Where students lose marks in Sharing in a Ratio, Mixtures and Unit Conversions

The slips examiners see most often, and what to write instead.

Mistake: Dividing by one term instead of the sum of the terms
The number of groups is m + n.
Wrong working
"840 in the ratio 3 : 4: 840 / 3 = 280 and 840 / 4 = 210."
Right working
$840 \div 7 = 120$ ; shares $360$ and $480$ .
Do this: Write "groups = 3 + 4 = 7" as the first line.
Usually costs: The whole answer.
Mistake: Changing both ingredients when only one is added
The ingredient that is not added keeps its quantity.
Wrong working
"Add x kg of rice flour: new ratio (20 + x) : (30 + x) = 1 : 1."
Right working
Wheat stays 30 kg, so $20 + x = 30$ and $x = 10$ .
Do this: Mark the unchanged ingredient before writing an equation.
Usually costs: 2 marks.
Mistake: Multiplying instead of dividing in a conversion
A bigger unit means a smaller number.
Wrong working
"2.5 acres = 2.5 x 2.471 = 6.18 hectares"
Right working
$2.5 \div 2.471 \approx 1.01$ hectares.
Do this: Predict whether the answer should be bigger or smaller first.
Usually costs: The answer.
Mistake: Shares that do not add up to the total
A wrong group size makes the shares inconsistent; adding them exposes it.
Wrong working
"840 in the ratio 3 : 4: shares 360 and 420."
Right working
The shares must add to 840: $360 + 480 = 840$ .
Do this: Add the shares as the last line of every sharing answer.
Usually costs: The answer mark.

Exam tips

Open with "groups = sum of the ratio terms" and "one group = total divided by groups".
Verify by adding all shares.
For mixtures, tabulate the ingredient amounts before and after.
In conversion questions, quote the factor used and say if it is rounded.
Give final answers with units (rupees, litres, square feet).

Practice questions on Sharing in a Ratio, Mixtures and Unit Conversions

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): Dividing 42 in the ratio 4 : 3 gives 24 and 18.
Reason (R): The 7 equal groups each have size 42 divided by 7, which is 6.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Compute 4 x 6 and 3 x 6.
Answer
(a). Group size $= 42 \div 7 = 6$ , shares $24$ and $18$ . R is the working that gives A.
Q2 · 2 marks · Very short answerBoard level
A 360 mL syrup is made from sugar solution and water in the ratio 2 : 7. How much of each does it contain?
Hint
Total groups are 9.
Answer
Groups $= 2 + 7 = 9$ , each $= 360 \div 9 = 40$ mL. Sugar solution $= 2 \times 40 = 80$ mL; water $= 7 \times 40 = 280$ mL.
Q3 · 3 marks · Short answerHarder board level
Using 1 metre = 3.281 feet and 1 square metre = 10.764 square feet, convert (i) a rope of 25 feet to metres, to two decimal places and (ii) a room of 12 ft by 10 ft to square metres, to two decimal places.
Hint
Divide to go from feet to metres.
Answer
(i) $25 \div 3.281 \approx 7.62$ m. (ii) Area $= 12 \times 10 = 120$ sq ft, and $120 \div 10.764 \approx 11.15$ sq m. (Check: $12 \div 3.281 \approx 3.657$ m and $10 \div 3.281 \approx 3.048$ m, product $\approx 11.15$ sq m.)
Q4 · 4 marks · Long answerHOTS / competency
A 24 L mixture of red and yellow paint is in the ratio 3 : 5. After 6 L of yellow is added, find the new ratio in simplest form. How much red paint must be added to this new mixture to bring the ratio back to 3 : 5?
Hint
Find red and yellow first; yellow stays 21 L in the last step.
Answer
Red $= \frac{3}{8} \times 24 = 9$ L, yellow $= 15$ L. After adding 6 L yellow: $9 : 21 = 3 : 7$ . For the ratio $3 : 5$ with yellow fixed at 21 L, red must be $21 \times \frac{3}{5} = 12.6$ L, so add $12.6 - 9 = 3.6$ L of red.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: three partners invest 60,000, 40,000 and 20,000 in a shop.
(i) Write the investment ratio in simplest form. (1 mark)
(ii) In year 1 the profit is 1,80,000. Find each share. (1 mark)
(iii) In year 2 the third partner adds another 20,000 so that the investments are 60,000, 40,000 and 40,000. Write the new ratio. (1 mark)
(iv) Year 2 profit is 2,10,000. Find each share. (1 mark)
Hint
Simplify the ratios by the HCF.
Answer
(i) $60 : 40 : 20 = 3 : 2 : 1$ . (ii) Groups $= 6$ , each $= 1, 80, 000 \div 6 = 30, 000$ ; shares $90, 000$ , $60, 000$ and $30, 000$ . (iii) $60 : 40 : 40 = 3 : 2 : 2$ . (iv) Groups $= 7$ , each $= 2, 10, 000 \div 7 = 30, 000$ ; shares $90, 000$ , $60, 000$ and $60, 000$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How many equal groups when dividing in ratio a : b : c?Show answer
a + b + c.
2.How are profits shared among partners?Show answer
In the ratio of their investments (for equal time).
3.How many square metres in a hectare?Show answer
10,000.
4.How many square feet in an acre?Show answer
43,560.
5.How many cubic centimetres in one millilitre?Show answer
One.
6.When one ingredient of a mixture is added, what stays constant?Show answer
The other ingredient.

Sharing in a Ratio, Mixtures and Unit Conversions: frequently asked questions

Can the groups method be used for three or more parts?

Yes. For a : b : c the total number of groups is a + b + c.

Why do two conversion routes give slightly different answers?

The conversion factors are rounded, so results can differ in later digits; report a sensible number of digits.

Is temperature conversion a multiplication like other units?

No. The chapter notes that conversion between Celsius and Fahrenheit is more complicated: 0 degrees Celsius is 32 degrees Fahrenheit.

What if investments are made for different times?

The chapter's examples assume equal time. If times differ, the ratio would need to account for time, which is beyond this chapter.