Question

One number is bigger than another by 406. If the bigger number is divided by the smaller one, you’ll get the quotient 3 and remainder 66. Find the numbers.

Answer 1

The larger number is 576, the smaller number is 170.

Let x = the larger number and y = the smaller number.

We know that the larger number is 406 larger than the smaller one; therefore
x = y + 406

We also know
$\frac{x}{y}=3\frac{66}{y}$

Substituting for x we have:
$\frac{y+406}{y}=3+\frac{66}{y}$

When we convert mixed numbers to improper fractions, we multiply the whole number by the denominator and add the numerator:
$\frac{y+406}{y}=\frac{3y+66}{y}$

Since the denominators are the same, we can just set the numerators equal
y+406=3y+66

We cannot have a variable on both sides, so we will subtract y from both sides:
y+406-y=3y+66-y
406=2y+66

Subtract 66 from both sides:
406-66=2y+66-66
340=2y

Divide both sides by 2:
340/2 = 2y/2
170=y

The smaller number is 170.  Since the larger number is 406 larger, then it is 170+406=576.