What a ratio is

We often com­pare two amounts. Who scored more? How much older is one child than another? One way to com­pare is to see how many times big­ger one amount is than the other. We do that by divid­ing. The answer is called a ratio, and you will use it in almost every les­son of this chap­ter.

Com­par­ing by divid­ing#

Two stu­dents wrote the same test. The first stu­dent got 2 marks. The sec­ond stu­dent got 4 marks.

Divide the first num­ber by the sec­ond:

$$\displaystyle \frac{2}{4} = \frac{1}{2}$$

We write this as $1 : 2$ and read it as "1 is to 2". It tells us that for every 1 mark of the first stu­dent, the sec­ond stu­dent has 2 marks. The sec­ond score is two times the first.

Here is another one. Find the ratio of 18 years to 24 years.

$$\displaystyle \frac{18}{24} = \frac{3}{4}$$

So the ratio is $3 : 4$.

The same kind, the same units#

We can only com­pare things of the same kind. Marks with marks. Years with years. We can­not com­pare 5 kg with 3 metres.

The units must also be the same. If one length is in mil­lime­tres and the other is in cen­time­tres, change one of them first.

Exam­ple 1#

What is the ratio of 18 mm to 3 cm?

The units are dif­fer­ent, so change cen­time­tres into mil­lime­tres. 1 cm = 10 mm, so 3 cm = 30 mm.

$$\displaystyle \frac{18}{30} = \frac{3}{5}$$

The ratio is $3 : 5$.

If you for­get to change the units, you get $18 : 3 = 6 : 1$. That says 18 mm is six times as long as 3 cm, which is wrong. A 3 cm line is longer than an 18 mm line.

What a ratio is#

Ratio. The ratio between two quan­ti­ties of the same kind and in the same units is found by divid­ing the first quan­tity by the sec­ond.

The ratio of $a$ to $b$ is writ­ten as $a : b$ or as $\displaystyle \frac{a}{b}$.

A ratio has no units. The units can­cel when we divide. So we write $3 : 5$, not "3 mm : 5 mm".

The terms of a ratio#

The two num­bers in a ratio are called its terms.

In the ratio $a : b$:

• $a$ is the first term. It is also called the antecedent.
• $b$ is the sec­ond term. It is also called the con­se­quent.

In $7 : 9$, the antecedent is 7 and the con­se­quent is 9.

Order mat­ters. The ratio $2 : 3$ is not the same as $3 : 2$. If Ravi has 2 pen­cils and Sita has 3, the ratio of Rav­i's pen­cils to Sita's is $2 : 3$. The ratio of Sita's pen­cils to Rav­i's is $3 : 2$.

The sim­plest form of a ratio#

A ratio $a : b$ is in its sim­plest form when the H.C.F. of $a$ and $b$ is 1. That means no num­ber other than 1 divides both terms.

Is $12 : 16$ in sim­plest form? No. Both 12 and 16 can be divided by 4.

$$\displaystyle \frac{12}{16} = \frac{4 \times 3}{4 \times 4} = \frac{3}{4}$$

So $3 : 4$ is the sim­plest form of $12 : 16$.

The quick­est way is to divide both terms by their H.C.F. The H.C.F. of 12 and 16 is 4. Then $12 \div 4 = 3$ and $16 \div 4 = 4$.

Equal ratios#

A ratio does not change when we mul­ti­ply both terms by the same num­ber. It also does not change when we divide both terms by the same num­ber. The num­ber must not be zero.

$$\displaystyle \frac{a}{b} = \frac{m \times a}{m \times b} \qquad \frac{a}{b} = \frac{a \div m}{b \div m}$$

So $2 : 3$, $4 : 6$, $6 : 9$ and $20 : 30$ are all equal ratios. Ratios like these are called equiv­a­lent ratios.

Exam­ple 2#

Find the ratio of 45 min­utes to 2 hours.

Change hours into min­utes. 1 hour = 60 min­utes, so 2 hours = 120 min­utes.

The ratio is $45 : 120$. The H.C.F. of 45 and 120 is 15.

$$\displaystyle \frac{45}{120} = \frac{45 \div 15}{120 \div 15} = \frac{3}{8}$$

The ratio is $3 : 8$.

Exam­ple 3#

Find the ratio of 750 g to 2 kg.

1 kg = 1000 g, so 2 kg = 2000 g.

$$\displaystyle \frac{750}{2000} = \frac{3}{8}$$

The H.C.F. of 750 and 2000 is 250, and $750 \div 250 = 3$, $2000 \div 250 = 8$. The ratio is $3 : 8$.

Exam­ple 4#

Find the ratio of ₹1.50 to 75 paise.

Change rupees into paise. ₹1 = 100 paise, so ₹1.50 = 150 paise.

$$\displaystyle \frac{150}{75} = \frac{2}{1}$$

The ratio is $2 : 1$. The first amount is twice the sec­ond.

Steps to find a ratio#

1. Check that both quan­ti­ties are of the same kind.
2. Write both in the same unit.
3. Divide the first by the sec­ond.
4. Can­cel by the H.C.F. to get the sim­plest form.

Prac­tice#

1. Write the ratio of 15 to 25 in sim­plest form.
2. Write $36 : 48$ in sim­plest form.
3. Find the ratio of 40 cm to 2 m.
4. Find the ratio of 20 min­utes to 1 hour.
5. Find the ratio of 600 g to 1.5 kg.
6. In the ratio $7 : 9$, name the antecedent and the con­se­quent.
7. Is $14 : 21$ in sim­plest form? If not, sim­plify it.
8. A class has 18 boys and 12 girls. Find the ratio of the num­ber of girls to the total num­ber of stu­dents.
9. Ravi is 1.2 m tall and his father is 180 cm tall. Find the ratio of Rav­i's height to his father's height.
10. Find the ratio of 90 paise to ₹3.

Answers#