Problemas matemáticos

Solving Word Problems with Percents

We can break word problems with percents into smaller steps!

The following graphic breaks down how to solve word problems with percents into $3$ simple steps. Following these steps will allow you have the confidence to tackle all future word problems with percents!

You can see that we started with the problem, identified what we knew, plugged in values, and solved the proportion. These same steps can be used for other word problems.

Let's do another example!

Step One: Identify Knowns and Unknown

KNOWNS: We know that the referees gave out $124$ yellow cards and $\boxed{4 \text{ red cards}}.$ This means that the total number of cards in the tournament is: $$\begin{align} \text{yellow cards + red cards}&=124+4\\ &=128 \end{align}$$ So the total number of cards at the tournament is $\boxed{128\text{ cards}}.$

UNKNOWNS: We want to determine the $\boxed{\text{percent of the total cards that were red}}.$ Let's call this unknown value $x.$

Step Two: Plug Values Into Proportion

We can use proportions to solve for our unknown percent, $x.$ All we have to do it plug our known and unknown values into the proportion, then solve for $x!$

First, let's plug our values into their correct spots. Here is our general set up for a proportion: $$\dfrac{\text{Part}}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ Where do our values go?

Well, we know that our unknown value, $x,$ is a percent. So we have: $$\dfrac{\text{Part}}{\text{Whole}} = \dfrac{x}{100}$$

Next, let's figure out where to substitute $4$ and $128$ into the proportion. We are trying to determine what percent of the total cards were red. We know that $4$ cards is the part and $128$ cards is the whole. (Usually, your total will be the whole.) So: $$\dfrac{4}{128} = \dfrac{x}{100}$$

Step Three: Solve the Proportion

$$\begin{align} \dfrac{4}{128} &= \dfrac{x}{100}\\ 128x &= 4\cdot 100\\ 128x &= 400\\ \dfrac{\cancel{128}x}{\cancel{128}} &=\dfrac{400}{128}\\ x&=\dfrac{400}{128}\\ x&=3.125 \end{align}$$

Here, we used cross-multiplication, but you can also solve using the multiplication property. Make sure you always follow the Multiplication and Division Properties of Equality!

So, red cards made up $3.125\%$ of the total number of cards given out at the soccer tournament!

(Videos) Percent Applications

by mathman1024

This video by mathaman1024 works through a percent word problem.

Summary

There are three steps you can take with any percentage-based word problem

1. Identify the knowns and unknowns. Some of the information will be given to us, but not all of it.
2. Plug the values into the proportion. We need to set up a proportion that incorporates the known values and the variables.
3. Solve the proportion. Once we have a set proportion, we have to use our algebra skills to isolate the variable.

The example problem is: $\$109{,}570{,}000$ of Lone Star College's $\$317{,}790{,}000$ budget comes from property taxes. What percent of their budget comes from property taxes?

Step 1: Identify the $\color{#29A603}{knowns}$ and $\color{#8925AE}{unknowns}$ from the problem

$\color{#29A603}{\$109{,}570{,}000}$ of Lone Star College's $\color{#29A603}{\$317{,}790{,}000}$ budget comes from property taxes. What $\color{#8925AE}{percent}$ of their budget comes from property taxes?

Step 2: Plug Values Into Proportion

$$\dfrac{\color{#29A603}{\text{Part}}}{\color{#29A603}{\text{Whole}}} = \dfrac{\color{#8925AE}{\text{Percent}}}{100}$$ $$\dfrac{\color{#29A603}{\text{\$109,570,000}}}{\color{#29A603}{\text{\$317,790,000}}} = \dfrac{\color{#8925AE}{\text{Percent}}}{100}$$

Step 3: Solve Proportion

Word Problems with Percents

You need to translate problems with percentages into math and then solve them. You can break down this process into three steps.

1. Identify the knowns and unknown from the problem.
2. Convert values given in the problems into proportions.
3. Solve the proportion.

Let’s try an example.

Example 1

Suppose $60\%$ of people in a math class have a dog. If the math class consist of $150$ people, how many people in the math class has a dog?

Step 1: Identify the knowns and unknown from the problem

Knowns: We know that $60\%$ of people in a math class have a dog. We also know that there are $150$ people in the math class.

Unknowns: We want to determine how many people in the math class have a dog. We can call this unknown value $x$.

Step 2: Convert values given in the problems into proportions

Next, we can plug in our known and unknown values into the proportion.

Remember, this is our general set up for a proportion: $$\dfrac{\text{Part}}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ We are solving for $x$, which is a part of the class that has a dog, so we can plug $x$ into “Part”. $$\dfrac{x}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ Next, we can substitute $60\%$ and $150$ into the proportion. Since $60\%$ is a percent, we can plug $60$ into ‘Percent”. $150$ is the total number of students in the class, so $150$ is the whole. $$\dfrac{x}{150} = \dfrac{60}{100}$$

Step 3: Solve the proportion using cross multiplication

$$\begin{align} \dfrac{x}{150} &= \dfrac{60}{100} \\ 100x &= 150 \cdot 60 \\ 100x &= 9{,}000 \\ x &= \dfrac{9{,}000}{100} \\ x &= 90 \\ \end{align}$$

There are $90$ students in the class of $150$ that have a pet dog.

Let’s try another example.

Example 2

Step 1: Identify the knowns and unknown from the problem

Knowns: We know that Nick bought $2$ shirts and a pair of jeans. Each shirt costs $\$50$, and the pair of jeans costs $\$75$. We first need to calculate how much he spent on his purchase, before the discount. $$(2 \cdot 50)+(1 \cdot 75)= 100 + 75 = 175$$ We also know that Nick has a coupon for $25\%$ off the entire purchase.

Unknowns: We want to determine how much Nick would pay after the discount. We can call this unknown value $x$.

Step 2: Convert values given in the problems into proportions

Next, we can plug in our known and unknown values into the proportion.

Remember, this is our general set up for a proportion: $$\dfrac{\text{Part}}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ We are solving for $x$, which is a part of the entire purchase after the discount, so we can plug $x$ into “Part”. $$\dfrac{x}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ Next, we can substitute $25\%$ and $175$ into the proportion.Since $25\%$ is a percent, we can plug $25$ into ‘Percent”. $175$ is the total number of students in the class, so $175$ is the whole. $$\dfrac{x}{175} = \dfrac{25}{100}$$

Step 3: Solve the proportion using cross multiplication.

$$\begin{align} \dfrac{x}{175} &= \dfrac{25}{100} \\ 100x &= 175 \cdot 25 \\ 100x &= 4{,}375 \\ x &= \dfrac{4{,}375}{100} \\ x &= 43.75 \\ \end{align}$$

$\$43.75$ represents $25\%$ of her purchase. This needs to be subtracted from her total amount to get the actual amount that Nick paid. $$175 - 43.75= 131.25$$ Nick paid $\$131.25$ for his purchase.

Example 3

You will see percentages as test scores, interest rates, or sales. To understand how to calculate them, make sure you understand the relationship between proportions and percents.

Step $1$: Identify the knowns and unknown from the problem

Knowns: We know that the sales tax is $7\%$. We also know that the computer costs $\$1{,}200$ before tax.

Unknowns: We want to determine the amount of sales tax that Ben has to pay. We can call this unknown value $x$.

Step $2$: Convert values given in the problems into proportions

Next, we can plug in our known and unknown values into the proportion.

Remember, this is our general set up for a percent proportion: $$\dfrac{\text{Part}}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ We are solving for $x$, which is the sales tax that Ben has to pay for the computer, so we can call this value “Part”. $$\dfrac{x}{\text{Whole}} = \dfrac{\text{Percent}}{100}$$ Next, we can substitute $7\%$ and $\$1{,}200$ into the proportion. Since $7\%$ is a percent, we can plug $7$ into ‘Percent”. $1{,}200$ is the price of the computer, so we can plug $1{,}200$ into “Whole” $$\dfrac{x}{1{,}200} = \dfrac{7}{100}$$

Step $3$: Solve the proportion using cross multiplication

$$\begin{align} \dfrac{x}{1{,}200} &= \dfrac{7}{100}\\ 100x &= 8{,}400 \\ x &= \dfrac{8{,}400}{100} \\ x&= 84\\ \end{align}$$

$\$84$ represents the sales tax that Ben has to pay. The total price that Ben would have to pay for the new computer is Price of Computer + Sales Tax, so the total price he has to pay is: $$\$1{,}200 + \$84 = \$1{,}284$$