# How do you find a number which exceeds 10 by as much as twice the number exceeds 38?

Dec 28, 2016

Write an equation: $2 x - 38 = x - 10$

Solve it to find $x = 28$

#### Explanation:

Let the unknown number be $x$

Twice the number exceeds (is more than) 38.
This difference can be written as: $2 x - 38$

The number exceeds 10.
This difference can be written as:$x - 10$

The two differences are equal to each other.

You can now write an equation showing this information.

$2 x - 38 = x - 10 \text{ } \leftarrow$ now solve the equation for $x$

$2 x - x = - 10 + 38$

$x = 28 \text{ } \leftarrow$ This is the number

Let's check:

$2 \times 28 - 38 = 18$

$28 - 10 = 18$

The differences are the same.

Dec 28, 2016

Depending upon how you translate the given statement,
the number is either $28$ or $66$.
(see below)

#### Explanation:

Let the number be represented by the variable $n$

There are two possible interpretations of the given statement:

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Interpretation 1

$\textcolor{red}{\text{a number which exceeds 10")color(green)(" by as much as")color(blue)(color(magenta)(" twice the number")" exceeds 38}}$

$\textcolor{red}{\text{a number which exceeds 10 ")color(green)(=)color(blue)(color(magenta)(" twice the number")" exceeds 38}}$

color(red)(n-10)color(green)(=)color(blue)(color(magenta)(" twice the number")" exceeds 38")

$\textcolor{red}{n - 10} \textcolor{g r e e n}{=} \textcolor{b l u e}{\textcolor{m a \ge n t a}{2 n} \text{ exceeds 38}}$

$\textcolor{red}{n - 10} \textcolor{g r e e n}{=} \textcolor{b l u e}{\textcolor{m a \ge n t a}{2 n} - 38}$

Subtract $n$ from both sides
$- 10 = n - 38$

Add $38$ to both sides
$28 = n \textcolor{w h i t e}{\text{XXXX}} \mathmr{and} n = 28$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Interpretation 2

$\textcolor{red}{\text{a number which exceeds 10")color(green)(" by as much as")color(blue)(color(magenta)(" twice")" the number exceeds 38}}$

$\textcolor{red}{\text{a number which exceeds 10 ")color(green)(=)color(blue)(color(magenta)(" twice")" the number exceeds 38}}$

color(red)(n-10)color(green)(=)color(blue)(color(magenta)(" twice")" the number exceeds 38")

$\textcolor{red}{n - 10} \textcolor{g r e e n}{=} \textcolor{b l u e}{\textcolor{m a \ge n t a}{\text{twice }} n - 38}$

color(red)(n-10)color(green)(=)color(blue)(color(magenta)2(n-38)

Expanding the right side
$\textcolor{red}{n - 10} = 2 n - 76$

Subtract $n$ from both sides
$- 10 = n - 76$

Add $76$ to both sides
$66 = n \textcolor{w h i t e}{\text{XXXX}} \mathmr{and} n = 66$