Unitary method and direct proportion

If 5 pens cost ₹30, what do 8 pens cost? Most peo­ple find the cost of one pen first, and then the cost of 8. That sim­ple idea has a name: the uni­tary method. In this les­son you will use it, and then meet direct pro­por­tion, which explains why it works.

The uni­tary method#

"Unit" means one. In the uni­tary method we first find the value of one thing. Then we mul­ti­ply to get the value of as many as we want.

So 8 pens cost ₹48.

Uni­tary method. A method in which we first find the value of one unit, and then use it to find the value of the quan­tity we need, is called the uni­tary method.

Exam­ple 1#

3 kg of rice can feed 20 peo­ple. How much rice is needed for 50 such peo­ple?

$$\displaystyle \frac{3}{20} \times 50 = \frac{150}{20} = 7.5$$

So 7.5 kg of rice is needed.

Direct pro­por­tion#

Look at the cost of sugar.

When the weight goes up, the cost goes up. When the weight goes down, the cost goes down. And it goes up or down in the same ratio: 3 times the sugar costs 3 times as much, and half the sugar costs half as much.

Divide each cost by its weight: $36 \div 1 = 36$, $108 \div 3 = 36$ and $\displaystyle 18 \div \frac{1}{2} = 36$. The answer is always the same.

Direct pro­por­tion. Two quan­ti­ties $x$ and $y$ are in direct pro­por­tion if, when one increases, the other increases in the same ratio, and when one decreases, the other decreases in the same ratio. This is also called direct vari­a­tion.

We write $x \propto y$ and read it as "$x$ is directly pro­por­tional to $y$".

When $x \propto y$, the ratio of the two stays the same:

$$\displaystyle x = ky \quad \text{or} \quad \frac{x}{y} = k$$

The num­ber $k$ is called the con­stant. It does not change.

So if $x_1$ and $y_1$ go together, and $x_2$ and $y_2$ go together, then

$$\displaystyle \frac{x_1}{y_1} = \frac{x_2}{y_2}$$

Cross mul­ti­ply, and you get the work­ing rule:

$$x_1 y_2 = x_2 y_1$$

In the sugar table, take $x_1 = 1$, $y_1 = 36$, $x_2 = 3$ and $y_2 = 108$. Then $x_1 y_2 = 1 \times 108 = 108$ and $x_2 y_1 = 3 \times 36 = 108$. They agree.

Solv­ing prob­lems with direct pro­por­tion#

Write the two quan­ti­ties in a table. Put the unknown as a let­ter. Then cross mul­ti­ply.

Exam­ple 2: Cloth#

5 metres of a cloth cost ₹210. What is the cost of 2 metres of the same cloth?

More cloth costs more, so this is direct pro­por­tion. Cross mul­ti­ply:

$$\displaystyle \begin{aligned} 5 \times y &= 2 \times 210 \\ y &= \frac{2 \times 210}{5} = 84 \end{aligned}$$

2 metres cost ₹84.

By the uni­tary method: 1 metre costs $210 \div 5 = 42$ rupees, so 2 metres cost $42 \times 2 = 84$ rupees. Same answer.

Exam­ple 3: A pole and a tree#

An elec­tric pole 14 m high casts a shadow 10 m long. At the same time, a tree casts a shadow 15 m long. How tall is the tree?

At the same time of day, a taller object casts a longer shadow, in the same ratio.

$$\displaystyle \begin{aligned} 14 \times 15 &= x \times 10 \\ x &= \frac{14 \times 15}{10} = 21 \end{aligned}$$

The tree is 21 m tall.

Exam­ple 4: Bananas#

4 dozen bananas cost ₹60. How many bananas can you buy for ₹22.50?

4 dozen $= 4 \times 12 = 48$ bananas.

$$\displaystyle \begin{aligned} x \times 60 &= 48 \times 22.50 \\ x &= \frac{48 \times 22.50}{60} = 18 \end{aligned}$$

You can buy 18 bananas. (One banana costs $60 \div 48 = 1.25$ rupees, and $18 \times 1.25 = 22.50$.)

Exam­ple 5: A pho­to­graph of a bac­terium#

A pho­to­graph of a bac­terium is enlarged 50,000 times. In the pho­to­graph it is 5 cm long. What is its real length?

$$\displaystyle \begin{aligned} x \times 50000 &= 1 \times 5 \\ x &= \frac{5}{50000} = \frac{1}{10000} \end{aligned}$$

The real length is $\displaystyle \frac{1}{10000}$ cm, which is 0.0001 cm. Using pow­ers of ten, $\displaystyle \frac{1}{10} = 10^{-1}$, $\displaystyle \frac{1}{100} = 10^{-2}$, $\displaystyle \frac{1}{1000} = 10^{-3}$, and so $\displaystyle \frac{1}{10000} = 10^{-4}$. The length is $10^{-4}$ cm.

How to spot direct pro­por­tion#

Ask your­self: if I dou­ble one quan­tity, does the other dou­ble too?

• Num­ber of pens and their cost: yes. Direct.
• Dis­tance trav­elled at a fixed speed and the time taken: yes. Direct.
• Num­ber of work­ers and the days they take to fin­ish a job: no. More work­ers take fewer days. This is a dif­fer­ent kind of pro­por­tion. You will meet it in the next les­son.

Prac­tice#

1. 6 note­books cost ₹90. Find the cost of 11 note­books.
2. A car trav­els 84 km on 7 litres of petrol. How far will it go on 12 litres?
3. 25 m of wire weighs 4 kg. What does 40 m of the same wire weigh?
4. 3 dozen eggs cost ₹216. Find the cost of 8 eggs.
5. A pole 6 m high casts a shadow 4 m long. At the same time a tower casts a shadow 24 m long. How tall is the tower?
6. On a map, 2 cm stands for 15 km. What dis­tance does 9 cm stand for?
7. 12 pen­cils cost ₹42. How many pen­cils can you buy for ₹77?
8. A model of a ship is made to the scale $1 : 200$. The model is 60 cm long. How long is the real ship, in metres?
9. $x$ and $y$ are in direct pro­por­tion. When $x = 8$, $y = 20$. Find $y$ when $x = 14$, and find $x$ when $y = 45$.

Answers#