# What number is 110% of 150?

Jun 13, 2018

165

#### Explanation:

Set up a proportion

$x$ = the 110% of 150 the part
150 = the total parts y
110% = the percent part
100% = the total percent.

Using the principle of ratio I choose to set 100 as the equivalent of the original quantity of 150 and 110 as the equivalent of the new unknown quantity.

Let $x$ be the new unknown quantity giving the ratio

$\left(\text{New quantity")/("original quantity}\right) = \frac{110}{100} = \frac{x}{150}$

Multiply both sided by 150

$\frac{110 \times 150}{100} = \frac{x \times 150}{150}$

This gives

$\frac{16500}{100} = x$

$165 = x$

Jun 13, 2018

165

Detailed explanation given. The actual maths is very short.

#### Explanation:

$\textcolor{b l u e}{\text{The underlying principle of percentage}}$

$\textcolor{red}{\text{It just a fraction.}}$ However, it is a special fraction because the bottom number (denominator) is fixed at 100.

The symbol % behaves like a unit of measurement but one that is worth $\times \frac{1}{100}$ including the multiply sign

So 110% is the same as $110 \times \frac{1}{100} = \frac{110}{100}$

The question states 110% color(red)(" of ") 150

Translating the word 'of' into mathematics it becomes multiply

$\textcolor{g r e e n}{110 \textcolor{w h i t e}{\text{dd")color(blue)(%)color(white)("dd.")color(red)("of")color(white)("dd}} 150}$

$\textcolor{g r e e n}{110 \textcolor{w h i t e}{\text{dd")color(blue)(%)color(white)("dd.")color(red)(xx)color(white)("d}} 150}$

$\textcolor{g r e e n}{110 \textcolor{b l u e}{\overbrace{\times \frac{1}{100}}} \textcolor{w h i t e}{\text{d")color(red)(xx)color(white)("d}} 150}$

Giving:

$\frac{110}{100} \times 150 \leftarrow \text{ Starting point}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{A trick to calculating this in the head}}$

$\frac{110}{100} \to \frac{100}{100} + \frac{10}{100} = 1 + \frac{1}{10}$

$\textcolor{w h i t e}{\text{d}}$

$\textcolor{w h i t e}{\text{d")1color(white)(".}} \times 150 = 150$
$\frac{1}{10} \times 150 = \underline{\textcolor{w h i t e}{1} 15 \leftarrow \text{ Add}}$
$\textcolor{w h i t e}{\text{dddddddddd}} 165$