3.06 Proportional reasoning with percents

Recall that a percent is a ratio where the denominator is $$100. Because of this definition, we can use proportional reasoning strategies to solve problems with percents.

Proportions can be used to represent percent problems as follows:

 

Worked example

Question 1

Evaluate: Use a proportion to answer the question, "What percent of 20 is 3?"

Think: We can translate the statement to a proportion. Then use proportional reasoning to solve for the unknown.

The percent is the unknown. So we can use the variable $$x to represent it.

The number $$3 is the part and $$20 is the whole.

Do:

$$percent100​	$$=	$$partwhole​
$$x100​	$$=	$$320​	$$x is the unknown percent. $$3 is the part. $$20 is the whole.
$$x100​	$$=	$$3×520×5​	Multiplying the fraction by $$55​ gives us a common denominator of $$100.
$$x100​	$$=	$$15100​
$$x	$$=	$$15	If the denominators in a proportion are the same, the numerators must also be the same.

 

So the number $$3 is $$15% of $$20. 

Reflect: Is there another method that we might use to check our solution?

 

Proportional reasoning with benchmark percents

Suppose we want to check our solution to the first worked example using a different method. Let's see how we can apply proportional reasoning to percents in a different way.

Worked example

QUESTION 2

Evaluate: Find $$15% of $$20.

Think: It might be easiest to find $$10% of $$20. 

We can then use half of that amount to find $$5% of $$20. If we add the two amounts, that will give us $$15% of $$20.

Do: First, find $$10% of $$20.

$$10% of $$20	$$=	$$0.10×20	Since $$10%=0.10
	$$=	$$2	Evaluate
$$5% of $$20	$$=	$$12​×2	Since $$5% is half of $$10%
	$$=	$$1
$$15%	$$=	$$10%+5%
	$$=	$$2+1
	$$=	$$3

 

So $$15% of $$20 is $$3.

Reflect: What other percents can we calculate using the benchmark of $$10%?

 

Practice questions

Question 3

Question 4

Question 5