Proportions and percent

A proportion is an equation that says that two or more ratios are equal. For instance if one package of cookies contain 20 cookies that would mean that 2 packages contain 40 cookies

$\\ \frac{20}{1}=\frac{40}{2} \\$

A proportion is read as "x is to y as a is to b".

$\\\frac{x}{y}=\frac{a}{b}\\\\\\ \frac{x}{{\color{red} y}}\cdot {\color{red} y}=\frac{a}{b}\cdot y \\\\\\ x\cdot b=\frac{a}{{\color{red} b}}\cdot y{\color{red} b}\\\\\\ xb=ay \\$

The products xb and ay are called cross products. The cross products of a proportion are always equal. If we want to check if two ratios form a proportion we can just check their cross products.

Example:

Use cross product to determine if the two ratios form a proportion.

$\\\frac{2}{16},\: \: \frac{5}{40}\\\\\\ \frac{2}{16}\overset{?}{=} \frac{5}{40}\\\\\\ \frac{2}{16}\cdot 16\cdot 40\overset{?}{=} \frac{5}{40}\cdot 16\cdot 40\\\\\\ \frac{2}{{\color{red} 16}}\cdot{\color{red} 16}\cdot 40\overset{?}{=} \frac{5}{{\color{red} 40}}\cdot 16\cdot {\color{red} 40}\\\\ 2\cdot 40\overset{?}{=}5\cdot 16 \\\\ 80=80 \\$

Here we can see that 2/16 and 5/40 are proportions since their cross products are equal.

Percent means hundredths or per hundred and is written with the symbol, %. Percent is a ratio were we compare numbers to 100 which means that 1% is 1/100.

Example:

In a box of eight donuts two have pink sprinkles. Try to express how many percent of the donuts in the box that have pink sprinkles using proportions.

$\\ \frac{2}{8}= \frac{x}{100}\\\\\\ \frac{2}{{\color{red} 8}}\cdot {\color{red} 8}= \frac{x}{100}\cdot 8\\\\\\ 2\cdot 100= \frac{8x}{{\color{red} 100}}\cdot {\color{red}100}\\\\\\ \frac{200}{8}=\frac{{\color{red} 8}x}{{\color{red} 8}}\\\\ x=25\% \\$

This proportion is called the percent proportion.

$\\\frac{a}{b}=\frac{x}{100}\\$

Fractions, percent and decimals can all represent the same number, but they are expressed differently.

Decimal è percent: We can convert decimals into percent by writing the decimal as a fraction with the denominator 1. We then extend the fraction by 100 to get the decimal in percent.

Example:

Write 0.27 in percent.

$\\0.27\rightarrow \frac{0.27}{1}\cdot 100=\frac{27}{100}\: \: or\: \: 27\%\\$

Percent è decimal: write the percent as a fraction with the denominator 100. A good thing to remember is that 100% = 1 since

$\\\frac{100}{100}=1\\$

Example:

Write 89% as a decimal.

$\\\frac{x}{100}=\frac{89}{100}=0.89\\$

Video lesson: Write in percent or as decimals

Next Class: Ratios and percent, Solving problems with percent

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