1.7 – Percents

Percents

There are three types of percent problems.

What is 30% of 250?
This is the first type of percent problem. We will want to begin by writing it as a mathematical equation: $\underbrace{\text{What }}_{x}\underbrace{\text{ is }}_{=} \underbrace{30\%}_{0.3}\underbrace{\text{ of }}_{\cdot} \underbrace{250}_{250}?$ $x=0.3\cdot250$ Notice that we change the percentage to a decimal in order to use it in an equation. We will typically do this any time we use a percent mathematically. We can solve this equation simply enough by multiplying: $x=75$

What is 30% of 250?

This is the first type of percent problem. We will want to begin by writing it as a mathematical equation:
$\underbrace{\text{What }}_{x}\underbrace{\text{ is }}_{=} \underbrace{30\%}_{0.3}\underbrace{\text{ of }}_{\cdot} \underbrace{250}_{250}?$
$x=0.3\cdot250$
Notice that we change the percentage to a decimal in order to use it in an equation. We will typically do this any time we use a percent mathematically. We can solve this equation simply enough by multiplying:
$x=75$

25 is what percent of 400?
This is the second type of percent problem. Let’s rewrite it in a mathematical equation: $\underbrace{25}_{25}\underbrace{\text{ is }}_{=}\underbrace{\text{ what percent }}_{x}\underbrace{\text{ of }}_{\ \cdot} \underbrace{400}_{400}?$ $25=x\cdot 400$ Here, we can divide each side of the equation by 400: $\frac{25}{400}=x$ And then solve by performing the division: $0.0625=x$ Notice here that when we solve for $x$ , we have a decimal. The question, however, asked for a percent, so we must convert this. We have $x=6.25\%$

25 is what percent of 400?

This is the second type of percent problem. Let’s rewrite it in a mathematical equation:
$\underbrace{25}_{25}\underbrace{\text{ is }}_{=}\underbrace{\text{ what percent }}_{x}\underbrace{\text{ of }}_{\ \cdot} \underbrace{400}_{400}?$
$25=x\cdot 400$
Here, we can divide each side of the equation by 400:
$\frac{25}{400}=x$
And then solve by performing the division:
$0.0625=x$
Notice here that when we solve for $x$ , we have a decimal. The question, however, asked for a percent, so we must convert this. We have $x=6.25\%$

8 is 12.5% of what number?
This is the third type of percent problem, translating into a mathematical equation gives: $8=0.125\cdot x$ Again, we have the percentage converted to a decimal. Solving by dividing each side by 0.125 gives: $\frac{8}{0.125}=\frac{\cancel{0.125}x}{\cancel{0.125}}$ $64=x$

8 is 12.5% of what number?

This is the third type of percent problem, translating into a mathematical equation gives:
$8=0.125\cdot x$ Again, we have the percentage converted to a decimal. Solving by dividing each side by 0.125 gives: $\frac{8}{0.125}=\frac{\cancel{0.125}x}{\cancel{0.125}}$

$64=x$

Discount and Markup

Discounts and Markups are an application of percents:

A retail store sells t-shirts for $50. If they go on sale for 25% off, what is the discount, and what is the new sales price?
To find the discount amount, we simply need to find the value that is 25% of $50. That is, we need to solve the equation: $x=0.25\cdot50$ Solving this equation simply involves multiplying the right hand side: $x=12.5$ So the discount is $12.50.To find the sale price then, we simply subtract the discount from the original price of the shirt. $\text{Sale price }= 50-12.5$ $\text{Sale price }= 37.5$ So the sales price is $37.50

A retail store sells t-shirts for $50. If they go on sale for 25% off, what is the discount, and what is the new sales price?

To find the discount amount, we simply need to find the value that is 25% of $50. That is, we need to solve the equation:
$x=0.25\cdot50$
Solving this equation simply involves multiplying the right hand side:
$x=12.5$
So the discount is $12.50.To find the sale price then, we simply subtract the discount from the original price of the shirt. $\text{Sale price }= 50-12.5$

$\text{Sale price }= 37.5$
So the sales price is $37.50

A retail store purchases jeans at a price of $12 per pair. They then markup the price of the jeans by 40%. What is the markup, and the selling price of the jeans?
The markup is found by multiplying the markup percent by the purchase price: $\text{Markup }=0.4\cdot12$ $\text{Markup }=4.8$ So the markup is $4.80The selling price of the jeans then is found by adding the markup to the purchase price. $\text{Selling price }=12+4.8$ $\text{Selling price }=16.8$ The jeans are sold for $16.80

A retail store purchases jeans at a price of $12 per pair. They then markup the price of the jeans by 40%. What is the markup, and the selling price of the jeans?

The markup is found by multiplying the markup percent by the purchase price:
$\text{Markup }=0.4\cdot12$
$\text{Markup }=4.8$
So the markup is $4.80The selling price of the jeans then is found by adding the markup to the purchase price. $\text{Selling price }=12+4.8$

$\text{Selling price }=16.8$
The jeans are sold for $16.80

Percent Increase and Percent Decrease

Percent increase is the measure of an increase, as a percentage of the original value.

Percent Increase
Percent Increase can be found by the formula $\text{\% Increase }=\frac{\text{New}-\text{Old}}{\text{Old}}\times100\%$

Jill pays rent of $400. Her landlord raises her rent to $420. What was her percent increase?
Jill’s new rent is $420, and her old rent was $400. Plugging these values into the formula yields: $\text{\% Increase }=\frac{420-400}{400}\times100\%$ $\text{\% Increase }=\frac{20}{400}\times100\%$ $\text{\% Increase }=0.05\times100\%$ $\text{\% Increase }=5\%$

Percent Decrease is a measure of decrease, as a percent of the original amount.

Percent Decrease
Percent decrease is calculated very similarly to percent increase. It is, however, slightly different. The formula for percent decrease is: $\text{\% Decrease }=\frac{\text{Old}-\text{New}}{\text{Old}}\times100\%$

Trent usually purchases $320 worth of chicken feed every month. One month, he only purchases $240 worth. What was the percent decrease in his purchase?
Notice that for percent decrease, only the order of the difference in the numerator has changed. – We can substitute the values into this formula and simplify: $\text{\% Decrease }=\frac{\text{320}-\text{240}}{\text{320}}\times100\%$ $\text{\% Decrease }=\frac{80}{320}\times100\%$ $\text{\% Decrease }=0.25\times100\%$ $\text{\% Decrease }=25\%$

PreCalculus I

1.7 – Percents

Percents

Discount and Markup

Percent Increase and Percent Decrease

Modal title