### Math Notes

Subjects

#### Algebra Solutions

##### Topics || Problems

The sum of the reciprocals of two numbers is 7. The larger reciprocal exceeds the smaller by $$\frac{7}{ 3}$$. Find the numbers. Solution:

Let $$x,y$$ be the numbers.

$$\frac{1}{x} + \frac{1}{y} = 7$$, let $$\frac{1}{x}$$ be the larger reciprocal.

$$\frac{1}{x} = \frac{1}{y} + \frac{7}{3}$$

$$\frac{1}{y} + \frac{7}{3} + \frac{1}{y} = 7$$

$$\frac{2}{y} = \frac{14}{3}$$

$$\frac{3}{7} = y$$

$$\frac{1}{x} =\frac{7}{3} + \frac{7}{3}$$

$$\frac{1}{x} = \frac{14}{3}$$

$$\frac{3}{14} = x$$

Thus the numbers are $$\frac{3}{7} and \frac{3}{14}$$