Direct Proportion

What is Direct Proportion?

Two quantities are said to be in direct proportion if their ratios always remain the same as the quantities increase or decrease. In other words, as one quantity increases, the other increases, and vice-versa, at the same rate.

Worked Example 11.2.1:

If 20 pens cost ₦100, what is the cost of 45 similar pens?

Solution

Let the cost of 45 pens be ₦x 

Express each quantity as a ratio:

Number of pens =  20 : 45

cost of pens = ₦100  

These ratios are equivalent for direct proportion

20  : 45    =  100 :   x

\( \frac{20}{45} = \frac{100}{x} \)

cross multiply

\( \scriptsize 20 \: \times \: x = 45 \: \times \: 100 \)

divide both sides by 20

\( \frac{20x}{20} = \frac{45 \: \times \: 100}{20} \)

x = 45 × 5  

=   ₦225

∴ 45 similar pens costs ₦225

Worked Example 11.2.2:

If 9000 g of meat cost ₦3150, what is the cost of \( \scriptsize 1 \normalsize \frac {1}{2}\scriptsize kg\)?

Solution

Let the cost of  \( \scriptsize 1 \normalsize \frac {1}{2}\scriptsize kg\) be x

Mass of meat   =  9000g :  \( \scriptsize 1 \normalsize \frac {1}{2}\scriptsize kg\)

= 9000g : \( \frac {3}{2}\scriptsize kg\)

convert kg to g

= 9000g : \( \frac {3}{2}\scriptsize \: \times \: 1000g\)

= 9000g : 1500g

= 90 : 15

= 6 : 1

Cost of meat = 3150 :  x

Both ratios are equivalent

6 : 1   =  3150 :  x

\( \frac{6}{1} = \frac{3150}{x} \\ \scriptsize cross \: multiply \\ \scriptsize 6x = 3150\\ \\ \scriptsize x = \normalsize \frac{3150}{6} \\ \\ \scriptsize x = ₦525\)