MathsCBSE Class 8 · Proportional Reasoning-2

Map Scales and Multi-term Ratios for Class 8

Q: What if the RF is given as 1 cm = 5 km?

Convert to a ratio: 5 km = 5,00,000 cm, so the RF is 1 : 5,00,000.

Q: Do triangles with sides in the ratio 3 : 4 : 5 have to be congruent?

No. They are scaled copies of each other; they are congruent only if the scale factor is the same.

Q: Can a ratio have decimal terms?

Yes, as in 1 : 1.5 : 3; you can also multiply all terms by 2 to get 2 : 3 : 6.

Q: How do I check a multi-term share?

Add the shares; the sum should equal the whole.

Updated 17 October 2026 · Fact-checked

A map scale (representative fraction, RF) such as 1 : 20,00,000 means 1 cm on the map is 20,00,000 cm = 20 km on the ground. A ratio can have more than two terms, e.g. 2 : 3 : 5; to divide x in the ratio a : b : c, find the value of one part x / (a + b + c) and multiply. Exams ask for map distances, mixtures, triangle angles and sides in a ratio, and area on maps.

Each 1 cm on the map is 20 km on the ground, so A to B at 3 cm is 3 × 20 = 60 km in a straight line.

Illustration drawn for QuestionBuddy.

Understand Map Scales and Multi-term Ratios

Proportion with two ratios. Two ratios $a : b$ and $c : d$ are proportional if $a \times d = b \times c$ (cross-multiplication). In the chapter, 6 cups of rice to 3 of urad dal ( $6 : 3$ ) and 4 cups to 2 ( $4 : 2$ ) give the cross products $6 \times 2 = 12$ and $3 \times 4 = 12$ , so they are proportional.

Ratios on maps. The Representative Fraction (RF), such as $1 : 60, 00, 000$ , is the ratio of a distance on the map to the corresponding geographical (straight-line) distance on the ground. It means 1 cm on the map stands for $60, 00, 000$ cm on the ground, which is $60$ km (divide by $1, 00, 000$ to convert cm to km). The distance by road is not the geographical distance.

*Ground distance* $=$ map distance $\times$ the second term of the RF (convert the unit at the end).
*RF* $=$ map distance $:$ ground distance, written with both in the same unit and simplified so the first term is 1.
Two maps of the same region at different scales should give about the same ground distance.
A scale applies to lengths. Areas on a map scale as the square of the length scale: if 1 cm is 5 km, then 1 sq cm is 25 sq km.

Ratios with more than two terms. A spice mix with coriander, chillies, toor dal and fenugreek in the ratio $8 : 4 : 2 : 1$ has a ratio of four terms. If one quantity changes by a factor, every other must change by the same factor to keep the mix: half the chillies means half of everything.

Dividing a whole in a multi-term ratio. To divide $x$ in the ratio $a : b : c$ :
1. Add the terms: $a + b + c$ is the number of equal parts.
2. One part $= \frac{x}{a + b + c}$ .
3. The shares are $a$ , $b$ and $c$ times one part.
In the chapter's concrete mix (cement : sand : gravel $= 1 : 1.5 : 3$ ), one batch has $1 + 1.5 + 3 = 5.5$ units. For 110 units there are $110 \div 5.5 = 20$ batches, giving 20, 30 and 60 units.

Ratios in geometry. The angles of a triangle add to $18 0^{\circ}$ , so angles in the ratio $1 : 3 : 5$ are $2 0^{\circ}, 6 0^{\circ}, 10 0^{\circ}$ . Triangles with sides in the ratio $3 : 4 : 5$ are scaled copies of one another (not congruent unless the scale is the same). A triangle with sides in the ratio $1 : 3 : 5$ cannot exist because $1 + 3 < 5$ (the two shorter sides must add to more than the longest side).

The pie charts and inverse proportion are on the next topic page of this chapter.

Map Scales and Multi-term Ratios: formulas to remember

Cross-multiplication test: $a : b :: c : d ⟺ a d = b c$; Proportional ratios.
Ground distance from the map: $ground distance = map distance \times n (RF 1 : n)$; Convert cm to km by dividing by 1,00,000.
RF from two distances: $RF = 1 : \frac{ground distance}{map distance}$; Same unit for both.
Area on a map: $map area = ground area \div n^{2} (RF 1 : n)$; Lengths scale by n, areas by n squared.
Dividing x in the ratio a : b : c: $\frac{a}{a + b + c} x, \frac{b}{a + b + c} x, \frac{c}{a + b + c} x$; One part = x divided by the sum of terms.
Triangle inequality for a ratio: $sum of two smallest terms > largest term$; Needed for sides in a ratio to form a triangle.

How to solve Map Scales and Multi-term Ratios questions

Follow this routine for map and multi-term ratio problems.

1For maps, write the RF as 1 : n and keep the map distance and ground distance in the same unit until the end.
2Multiply the map distance by n to get the ground distance; then convert (100 cm = 1 m, 1,00,000 cm = 1 km).
3To find the RF, divide the ground distance by the map distance and write 1 : (the quotient).
4For a ratio with several terms, add the terms to count the parts and divide the whole to find the value of one part.
5Multiply one part by each term to get the shares; add them to check they equal the whole.
6When one quantity is given, find the value of one part from it and scale the rest.
7For triangle problems, check the angle sum (180 degrees) or the inequality on sides.

Map Scales and Multi-term Ratios 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.

Reading a map

Suppose a map has the scale 1 : 25,00,000 and two towns are 6.4 cm apart on it.

Find the geographical distance in km.

Try it first, then show the solutionSolution

Ground distance $= 6.4 \times 25, 00, 000 = 1, 60, 00, 000$ cm.
Convert: $1, 60, 00, 000 \div 1, 00, 000 = 160$ km.

Answer: 160 km

Memory hook: Multiply by the big term, then divide by 1,00,000 for km.

Mixing in a three-term ratio

Imagine a builder who mixes cement, sand and gravel in the ratio 1 : 1.5 : 3 for a beam.

She has 24 bags of sand. How many bags of cement and gravel does she need, and how many bags is the mix in all?

Try it first, then show the solutionSolution

Sand is 1.5 parts, so one part $= 24 \div 1.5 = 16$ bags.
Cement $= 16$ bags, gravel $= 3 \times 16 = 48$ bags.
Total $= 16 + 24 + 48 = 88$ bags.

Answer: 16 bags of cement, 48 of gravel, 88 bags in all

Memory hook: Find the value of one part from the quantity you know.

Tricks to remember Map Scales and Multi-term Ratios

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

Map times the second term
For RF 1 : n, multiply the map length by n to get the ground length in the same unit.
Use it when: Every map-distance question.
Lakh centimetres make a kilometre
1,00,000 cm = 1 km, so divide the ground distance in cm by 1,00,000.
Use it when: Converting map answers to km.
Add the parts first
For sharing, total parts = sum of the ratio terms; one part = whole divided by that sum.
Use it when: Multi-term sharing.
Areas use the square
If a length shrinks by n, an area shrinks by n squared.
Use it when: Area on maps.

Map Scales and Multi-term Ratios: solved examples

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

Example 1: Meaning of the scale · 1 mark · MCQBoard level

On a map with RF 1 : 50,000, 1 cm represents: (a) 50 m (b) 500 m (c) 5 km (d) 50 km

Try it first, then show the solutionSolution

1 cm on the map is 50,000 cm on the ground $= 500$ m (since 100 cm = 1 m).

Answer: (b) 500 m

Example 2: Distance from a map · 2 marks · Short answerBoard level

Two cities are 6.4 cm apart on a map of scale 1 : 25,00,000. Find the actual (geographical) distance in kilometres.

Try it first, then show the solutionSolution

Ground distance $= 6.4 \times 25, 00, 000$ cm $= 1, 60, 00, 000$ cm. (1 mark)
In km: $1, 60, 00, 000 \div 1, 00, 000 = 160$ km. (1 mark)

Answer: 160 km

Example 3: Finding the scale · 3 marks · Short answerHarder board level

Two towns are 144 km apart and are shown 9.6 cm apart on a map. Find the RF of the map.

Try it first, then show the solutionSolution

$144$ km $= 144 \times 1, 00, 000 = 1, 44, 00, 000$ cm. (1 mark)
Scale: $9.6 : 1, 44, 00, 000 = 1 : \frac{1 , 44 , 00 , 000}{9.6}$ . (1 mark)
$\frac{1 , 44 , 00 , 000}{9.6} = 15, 00, 000$ , so the RF is $1 : 15, 00, 000$ . (1 mark)

Answer: RF = 1 : 15,00,000

Example 4: Concrete in a decimal ratio · 3 marks · Short answerHarder board level

Cement, sand and gravel are mixed in the ratio 1 : 1.5 : 3 to make 187 units of concrete. Find the units of each.

Try it first, then show the solutionSolution

One batch has $1 + 1.5 + 3 = 5.5$ units. (1 mark)
Number of batches $= 187 \div 5.5 = 34$ . (1 mark)
Cement $= 34$ , sand $= 1.5 \times 34 = 51$ , gravel $= 3 \times 34 = 102$ ; check $34 + 51 + 102 = 187$ . (1 mark)

Answer: Cement 34, sand 51, gravel 102 units

Example 5: A triangle in the ratio 3 : 4 : 5 · 4 marks · Long answerHarder board level

A triangle has sides in the ratio 3 : 4 : 5 and perimeter 84 cm. Find the sides, show that the triangle is right-angled and find its area. Can a triangle have sides in the ratio 2 : 3 : 6?

Try it first, then show the solutionSolution

One part $= 84 \div 12 = 7$ , so the sides are 21, 28 and 35 cm. (1 mark)
$2 1^{2} + 2 8^{2} = 441 + 784 = 1225 = 3 5^{2}$ , so it is right-angled with legs 21 and 28 (the converse of the Baudhayana-Pythagoras theorem). (1 mark)
Area $= \frac{1}{2} \times 21 \times 28 = 294$ sq cm. (1 mark)
A triangle in the ratio 2 : 3 : 6 is impossible because $2 + 3 = 5 < 6$ . (1 mark)

Answer: Sides 21, 28, 35 cm; right-angled; area 294 sq cm; 2 : 3 : 6 cannot form a triangle.

Where students lose marks in Map Scales and Multi-term Ratios

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

Mistake: Not converting the unit of the ground distance
The product is in cm; the answer must be in km if asked.
Wrong working
"6.4 x 25,00,000 = 1,60,00,000 km"
Right working
$1, 60, 00, 000$ cm $= 160$ km.
Do this: Write the unit after every number.
Usually costs: The answer.
Mistake: Treating an area scale like a length scale
Areas scale as the square of the length scale.
Wrong working
"1 cm = 5 km, so 3 sq cm = 15 sq km."
Right working
$1 sq cm = 25$ sq km, so 3 sq cm is 75 sq km.
Do this: Square the length scale for areas.
Usually costs: 2 marks.
Mistake: Dividing by the first term instead of the sum of terms
The number of equal parts is the sum of all terms.
Wrong working
"187 in 1 : 1.5 : 3: 187 / 1 = 187 units of cement."
Right working
$187 \div 5.5 = 34$ per part.
Do this: Write "parts = sum of terms" first.
Usually costs: The whole part.
Mistake: Not checking that sides in a ratio form a triangle
The sum of the two shorter sides must exceed the longest.
Wrong working
"A triangle with sides in ratio 1 : 3 : 5 exists."
Right working
$1 + 3 < 5$ , so no triangle exists.
Do this: Apply the inequality before any construction or calculation.
Usually costs: 1 mark.

Exam tips

Write the RF as 1 : n and keep all lengths in centimetres until the end.
Check the final unit and magnitude: a map distance of a few cm on a country map means hundreds of km.
Add the shares at the end to match the total.
For area on maps, square the scale; write the length scale first, then its square.
For triangles, check the inequality or the angle sum before constructing.

Practice questions on Map Scales and Multi-term Ratios

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): On a map with RF 1 : 20,00,000, 1 cm represents 20 km.
Reason (R): 20,00,000 cm = 20 km.
(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
Divide by 1,00,000 to convert cm to km.
Answer
(a). $20, 00, 000 \div 1, 00, 000 = 20$ , so 1 cm stands for 20 km, and R is the conversion that proves it.
Q2 · 2 marks · Very short answerBoard level
A dye is made by mixing red, yellow, blue and white in the ratio 5 : 3 : 2 : 1. If 6 litres of blue are used, how much of the other colours are needed, and what is the total?
Hint
Blue is 2 parts.
Answer
One part $= 6 \div 2 = 3$ L. Red $= 15$ L, yellow $= 9$ L, white $= 3$ L. Total $= 15 + 9 + 6 + 3 = 33$ L.
Q3 · 3 marks · Short answerHarder board level
A purse has 100 coins of Rs 1, Rs 2 and Rs 5 in the ratio 5 : 3 : 2. How many coins of each kind are there and what is the total value?
Hint
Parts = 10; one part = 10 coins.
Answer
Parts $= 5 + 3 + 2 = 10$ , one part $= 100 \div 10 = 10$ coins. Rs 1 coins: 50, Rs 2 coins: 30, Rs 5 coins: 20. Value $= 50 \times 1 + 30 \times 2 + 20 \times 5 = 50 + 60 + 100 = 210$ rupees.
Q4 · 4 marks · Case-basedHOTS / competency
Case study: two maps show the same two cities. On map A, with RF 1 : 5,00,000, they are 8.4 cm apart; on map B they are 2.1 cm apart. A lake appears with area 3 sq cm on map A.
(i) Find the actual distance in km. (1 mark)
(ii) Find the RF of map B. (1 mark)
(iii) Find the actual area of the lake. (1 mark)
(iv) Find the area of the lake on map B. (1 mark)
Hint
Areas scale by the square of the length scale. 1 cm on map A is 5 km.
Answer
(i) $8.4 \times 5, 00, 000 = 42, 00, 000$ cm $= 42$ km. (ii) $42, 00, 000 \div 2.1 = 20, 00, 000$ , so RF $= 1 : 20, 00, 000$ . (iii) 1 sq cm on map A is $5 \times 5 = 25$ sq km, so the lake is $3 \times 25 = 75$ sq km. (iv) On map B, 1 cm is 20 km, so 1 sq cm is 400 sq km, and the lake is $75 \div 400 = 0.1875$ sq cm.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: a builder mixes cement, sand and gravel in the ratio 1 : 1.5 : 3.
(i) How many units are in one batch? (1 mark)
(ii) She has 24 bags of sand. How many bags of cement and gravel are needed and how many bags in all? (1 mark)
(iii) For 330 bags in all, find the number of bags of each. (1 mark)
(iv) What fraction of the mix is gravel? (1 mark)
Hint
One batch is 5.5 units.
Answer
(i) $1 + 1.5 + 3 = 5.5$ . (ii) One part $= 24 \div 1.5 = 16$ ; cement 16, gravel 48, total $16 + 24 + 48 = 88$ bags. (iii) $330 \div 5.5 = 60$ batches: cement 60, sand 90, gravel 180 bags. (iv) $\frac{3}{5.5} = \frac{6}{11}$ .

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.What does RF 1 : n mean?Show answer
1 cm on the map is n cm on the ground.
2.How many centimetres make a kilometre?Show answer
1,00,000.
3.How many parts does a : b : c have?Show answer
a + b + c.
4.How do areas scale on a map with RF 1 : n?Show answer
By n squared.
5.Can sides in the ratio 1 : 3 : 5 form a triangle?Show answer
No, since 1 + 3 is less than 5.
6.Is the map distance the same as the road distance?Show answer
No; it gives the geographical (straight-line) distance.

Map Scales and Multi-term Ratios: frequently asked questions

What if the RF is given as 1 cm = 5 km?

Convert to a ratio: 5 km = 5,00,000 cm, so the RF is 1 : 5,00,000.

Do triangles with sides in the ratio 3 : 4 : 5 have to be congruent?

No. They are scaled copies of each other; they are congruent only if the scale factor is the same.

Can a ratio have decimal terms?

Yes, as in 1 : 1.5 : 3; you can also multiply all terms by 2 to get 2 : 3 : 6.

How do I check a multi-term share?

Add the shares; the sum should equal the whole.