Direct Proportion

Two quantities a and b are said to be in direct proportion if

  • Increase in a increases the b
  • Decrease in a decreases the b

But the ratio of their respective values must be the same.

  • a and b will be in direct proportion if  \mathrm{\dfrac{a}{b}} = k(k is constant) or a = kb.
  • In such a case if b1, b2 are the values of b corresponding to the values a1, a2 of a respectively then, \mathrm{\dfrac{a_1}{b_2}=\dfrac{a_2}{b_1}}

Symbol of Proportion

When two quantities a and b are in proportion then they are written as a \propto b where \propto represents “is proportion to”.

Methods to solve Direct Proportion Problems

There are two methods to solve the problems related to direct proportion-

1. Tabular Method

As we know that

\mathrm{\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=\dfrac{a_3}{b_3}=\dfrac{a_n}{b_n}}

So, if one ratio is given then we can find the other values also. (The ratio remains the constant in the direct proportion).

Example

The cost of 4-litre milk is 200 Rs. Tabulate the cost of 2, 3, 5, 8 litres of milk of same quality.

Solution:

Let X litre of milk is of cost Y Rs.

X(Liter)23458
Y(Rupees)Y2Y3200Y4Y5

We know that as the litre will increase the cost will also increase and if the litre will decrease then the cost will also decrease.

Given,

 \dfrac{X_{1}}{Y_{1}}=\dfrac{4}{200}
 \text{a}) \dfrac{X_{1}}{Y_{1}}=\dfrac{X_{2}}{Y_{2}}
 \dfrac{4}{200}=\dfrac{2}{Y_2}
 4 Y_2=2 \times 200
 Y_2=\dfrac{2 \times 200}{4}
Y_2=100

So the cost of 2 litre milk is Rs.100

 \text{b})\ \dfrac{X_1}{Y_1}=\dfrac{X_3}{Y_3}
 \dfrac{4}{200}=\dfrac{3}{Y_3}
 4 Y_3=3 \times 200
 Y_3=\dfrac{3 \times 200}{4}
 Y_3=150

So the cost of 3 ltr milk is Rs. 150

 \text{c})\ \dfrac{X_1}{Y_1}=\dfrac{X_4}{Y_4}
 \dfrac{4}{200}=\dfrac{5}{Y_4}
 4 Y_4=5 \times 200
 Y_4=\dfrac{5 \times 200}{4}
 Y_4=250

So the cost of 5 ltr milk is Rs. 250

 \text d)\ \dfrac{X_{1}}{Y_{1}}=\dfrac{X_{5}}{Y_{5}}
 \dfrac{4}{200}=\dfrac{8}{Y_5}
 4 Y_5=8 \times 200
 Y_5=\dfrac{8 \times 200}{4}
Y_5=400

So the cost of 8 litre milk is Rs. 400

2. Unitary Method

If two quantities a and b are in direct proportion then the relation will be

\mathrm{k=\dfrac{a}{b}\ or\ a=k}

We can use this relation in solving the problem.

Example

If a worker gets 2000 Rs. to work for 4 hours then how much time will they work to get 60000 Rs.?

Solution:

Here, 

\mathrm{k=\dfrac{No.\ of\ hours}{Salary\ of\ worker}=\dfrac{4}{2000}=\dfrac{1}{500}}

\mathrm{a=\dfrac{1}{500}\times6000=12}

Hence, they have to work for 12 hours to get Rs. 60000

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