# The units digit of the two digit integer is 3 more than the tens digit. The ratio of the product of the digits to the integer is 1/2. How do you find this integer?

Aug 14, 2017

$36$

#### Explanation:

Suppose the tens digit is $t$.

Then the units digit is $t + 3$

The product of the digits is $t \left(t + 3\right) = {t}^{2} + 3 t$

The integer itself is $10 t + \left(t + 3\right) = 11 t + 3$

From what we are told:

${t}^{2} + 3 t = \frac{1}{2} \left(11 t + 3\right)$

So:

$2 {t}^{2} + 6 t = 11 t + 3$

So:

$0 = 2 {t}^{2} - 5 t - 3 = \left(t - 3\right) \left(2 t + 1\right)$

That is:

$t = 3 \text{ }$ or $\text{ } t = - \frac{1}{2}$

Since $t$ is supposed to be a positive integer less than $10$, the only valid solution has $t = 3$.

Then the integer itself is:

$36$