A Centimetre on Paper Can Stand for Five Kilometres

Because a map is drawn to a scale — a fixed proportion between paper and ground. If $1$ cm represents $5$ km, then $7$ cm represents

$$7\times 5=35\text{ km}$$

Every distance on the sheet is shrunk by the same factor, which is what keeps the map usable. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: testing direct proportion, map scales, three-term ratios, and sharing among three or more.

Divide each pair of corresponding values and check that the quotient stays constant. That constant is the factor linking them.

Worked example. Petrol used against distance covered:

- $2$ litres, $30$ km
- $4$ litres, $60$ km
- $6$ litres, $90$ km
- $8$ litres, $120$ km

Dividing distance by petrol each time:

$$\frac{30}{2}=15,\frac{60}{4}=15,\frac{90}{6}=15,\frac{120}{8}=15$$

The quotient is constant at 15 km per litre, so the two are directly proportional, and the relationship is $\text{distance}=15\times \text{petrol}$.

Worked example that fails. Hours worked against charge:

- $1$ hour, ₹5
- $2$ hours, ₹9
- $3$ hours, ₹13

$$\frac{5}{1}=5,\frac{9}{2}=4.5,\frac{13}{3}\approx 4.33$$

The quotient changes, so these are not proportional. What is constant here is the difference — ₹4 more per hour after a fixed ₹1 — so the relationship is additive.

In direct proportion, doubling one doubles the other. Going from 4 litres to 8 litres doubled the distance from 60 km to 120 km.

The test is the quotient, not the appearance of regularity, and that is the trap. The second table rises by a steady ₹4 each hour and looks orderly, yet doubling the hours from 1 to 2 does not double the charge from ₹5 to ₹10 — so a constant difference is never proportionality.

A scale states what one unit on the map represents on the ground, and converting is a single multiplication or division.

- Map to ground — multiply by the scale factor
- Ground to map — divide by it

Worked example. A map has the scale $1$ cm $=5$ km.

$$7\text{ cm on the map}\to 7\times 5=35\text{ km}$$

$$60\text{ km on the ground}\to \frac{60}{5}=12\text{ cm on the map}$$

$$2.5\text{ cm}\to 12.5\text{ km}$$

Worked example with a ratio scale. A scale written $1:50000$ means $1$ cm on the map represents $50000$ cm on the ground. Converting that:

$$50000\text{ cm}=500\text{ m}=0.5\text{ km}$$

So $4$ cm on this map represents $4\times 0.5=2$ km.

Worked example. On a scale of $1:200000$, one centimetre is $2$ km, so a road measuring $6.5$ cm is $13$ km long.

A house plan works the same way with a larger scale, such as $1$ cm $=1$ m.

The decision to make before calculating is the direction. Going from map to ground the number gets bigger, and going from ground to map it gets smaller — so an "actual distance" that comes out smaller than the map measurement means the multiplication and division have been swapped.

A ratio such as $a:b:c$ compares three quantities at once, and it is simplified by dividing all three terms by their HCF.

$$12:18:24=2:3:4$$

Finding a missing term from one known value. Three quantities are in the ratio $2:3:5$, and the middle one is $18$.

The middle term is $3$ parts, so

$$1\text{ part}=\frac{18}{3}=6$$

Therefore the three quantities are $2\times 6=12$, $18$, and $5\times 6=30$, making a total of $60$.

Combining two ratios into one. If $A:B=2:3$ and $B:C=4:5$, make $B$ match in both. The LCM of 3 and 4 is 12, so multiply the first ratio by 4 and the second by 3:

$$A:B=8:12,B:C=12:15$$

$$\text{so }A:B:C=8:12:15$$

Another: if $P:Q=3:4$ and $Q:R=8:9$, the LCM of 4 and 8 is 8, giving $P:Q=6:8$ and $Q:R=8:9$, so $P:Q:R=6:8:9$.

The step that makes combining work is matching the common term, and it cannot be skipped. Writing $A:B:C=2:3:5$ straight from $2:3$ and $4:5$ is wrong, because the $B$ in the first ratio counts 3 parts while the $B$ in the second counts 4 — they must be brought to the same number of parts before the three can sit in one statement.

Add all the ratio terms to get the total parts, find the value of one part, then multiply.

Worked example. Divide ₹2400 among three people in the ratio $3:4:5$.

Total parts $=3+4+5=12$, so one part $=\frac{2400}{12}=₹200$. The shares are

$$3\times 200=₹600,4\times 200=₹800,5\times 200=₹1000$$

Check: $600+800+1000=₹2400$, and $600:800:1000$ reduces to $3:4:5$. Both conditions hold.

Using share fractions gives the same result:

$$\frac{3}{12}\times 2400=600,\frac{4}{12}\times 2400=800,\frac{5}{12}\times 2400=1000$$

Worked example. Divide $360$ sweets among three children in the ratio $2:3:4$. Total parts $=9$, one part $=40$, so the shares are $80$, $120$ and $160$, adding to $360$.

Four shares. Divide $500$ g of a mixture in the ratio $1:2:3:4$. Total parts $=10$, one part $=50$ g, giving $50$, $100$, $150$ and $200$ g.

From one share back to the whole. If the largest share of a $3:4:5$ division is ₹1000, one part is $\frac{1000}{5}=₹200$, so the whole is $12\times 200=₹2400$.

The denominator must be the total parts, and the sum check is what catches an error. Using $\frac{3}{4}$ instead of $\frac{3}{12}$ for the first share would give ₹1800, and the three shares would then add to far more than the money available.