Sure, let's address each part of the problem step by step.
(i) Express the smaller number in terms of [tex]\( x \)[/tex]:
Given:
- The product of two numbers is 80.
- The greater number is [tex]\( x \)[/tex].
- The difference between the two numbers is 2.
Let the smaller number be [tex]\( y \)[/tex].
We know the product of the two numbers:
[tex]\[ xy = 80 \][/tex]
Thus, the smaller number [tex]\( y \)[/tex] can be expressed as:
[tex]\[ y = \frac{80}{x} \][/tex]
(ii) Find the numbers:
Next, we need to find the values of these numbers such that the difference between the greater and smaller number is 2.
Given:
[tex]\[ x - y = 2 \][/tex]
Substitute [tex]\( y \)[/tex] from step (i):
[tex]\[ x - \frac{80}{x} = 2 \][/tex]
To solve this equation for [tex]\( x \)[/tex], we'll rearrange it as:
[tex]\[ x^2 - 2x - 80 = 0 \][/tex]
This is a quadratic equation in the form of [tex]\( ax^2 + bx + c = 0 \)[/tex].
Solve this quadratic equation for [tex]\( x \)[/tex]:
[tex]\[ x = 10 \quad \text{(The positive solution since only positive solutions are relevant here)} \][/tex]
Now, find the value of the smaller number [tex]\( y \)[/tex]:
[tex]\[ y = \frac{80}{x} = \frac{80}{10} = 8 \][/tex]
Thus, the two numbers are:
- Greater number [tex]\( x = 10 \)[/tex]
- Smaller number [tex]\( y = 8 \)[/tex]
(iii) Calculate the percent by which the smaller number is less than the greater number:
We need to determine what percent the difference between the greater and smaller number is of the greater number.
Difference between the numbers:
[tex]\[ x - y = 10 - 8 = 2 \][/tex]
The percentage difference:
[tex]\[ \text{Percent Difference} = \left( \frac{x - y}{x} \right) \times 100 \][/tex]
Substitute the values:
[tex]\[ \text{Percent Difference} = \left( \frac{2}{10} \right) \times 100 = 20\% \][/tex]
Thus, the smaller number is 20% less than the greater number.
