Word Problems with Large & Small Numbers

Worked Example 8.4.1:

Given that x = 85 000, and y = 25 000 find the value of

a. \( \scriptsize x \: + \: y \)

b. \( \scriptsize x \: \times \: y \)

c. \( \scriptsize x \: \div \: y \)

Give your answer in standard form.

Solution

a. x  + y = \( \scriptsize 85\:000 \: + \: 25\:000\\ \scriptsize = 170\:000 \\ \scriptsize = 1.7 \: \times \: 10^5 \)

b. x ×  y   = \( \scriptsize 85\:000 \: \: \times \: \: 25\:000\\ \scriptsize = 8.5 \: \times \: 10^4 \: \times \: 2.5 \: \times \: 10^4 \\ \scriptsize = 21.25 \: \times \: 10^4 \: \times \: 10^4 \\ \scriptsize = 21.25 \: \times \: 10^{4\:+\:4}\\ \scriptsize = 21.25 \: \times \: 10^{8}\\ \scriptsize but \: 21.25 = 2.125 \: \times \: 10^1 \\ \scriptsize =\left( 2.125 \: \times \: 10^1\right) \: \times \: 10^8 \\ \scriptsize = 2.125 \: \times \: 10^{1\:+\:8}\\ \scriptsize = 2.125 \: \times \: 10^9 \)

 c. x ÷ y   = \( \frac{x}{y} \\= \frac{85\not{0}\not{0}\not{0}}{25 \not{0}\not{0}\not{0}} \\= \frac{85}{25} \\= \frac{17}{5} \\ = \scriptsize 3.4 \\ = \scriptsize 3.4 \: \times \: 10^0\)  

Worked Example 8.4.2:

The population of Nigeria is about 145 million and the population of Britain is about 58 million. Write these numbers in standard form and find the difference in population between the two countries.

Solution

Nigeria = 145 million

= 145 × 1 000 000

= 145  × 10⁶

but 145 = 1.45 × 10²

Therefore,

= (1.45 × 10²) × 10⁶

= 1.45 × 10²⁺⁶

= 1.45 × 10⁸

Britain = 58  million

= 58 × 1 000 000

= 58 × 10⁶

but 58 = 5.8 × 10¹

Therefore,

= (5.8 × 10¹) × 10⁶

= 5.8 × 10¹⁺⁶   

=  5.8 × 10⁷

Difference in population  =  145 million  – 58 million 

= 87 million

= 87 × 1 000 000

= 87 × 10⁶

but 87 = 87 × 10¹

Therefore,

= (8.7 × 10¹) × 10⁶ 

= 8.7 × 10¹⁺⁶

= 8.7 × 10⁷