The cooking club had 50 students. Then they gained 6 students.
This is a 12% increase, because \(6 \div 50 = 0.12\).
They now have 56 students, which is 112% of the starting amount.
\(1.12 \boldcdot 50 = 56\)
Unit 6 Family Materials
Percentage Increase and Decrease
Proportional Relationships with Fractions
This week your student is learning about proportional relationships that involve fractions and decimals. For example, a baker decides to start using \(\frac16\) less than the amount of sugar called for in each recipe. If the recipe calls for 2 cups of sugar, the baker will leave out \(\frac16 \boldcdot 2\), or \(\frac13\) cup of sugar. That means the baker will only use \(2 - \frac13\), or \(1\frac23\) cups of sugar.
| amount of sugar in the recipe (\(x\)) | amount of sugar the baker uses (\(y\)) |
|---|---|
| 1 cup | \(\frac56\) cup |
| \(1\frac12\) cups | \(1\frac14\) cups |
| 2 cups | \(1\frac23\) cups |
The amount of sugar the baker actually uses, \(y\), is proportional to the amount of sugar called for in the recipe, \(x\). The constant of proportionality is \(\frac56\).
\(y = x - \frac16 x\)
\(y = (1 - \frac16) x\)
\(y = \frac56 x\)
Another way to write this equation is \(y = 0.8\overline{3}x\). The line above the 3 tells us that if we use long division to divide \(5 \div 6\), we will keep getting the answer 3 over and over. This is an example of a repeating decimal.
Here is a task to try with your student:
The baker also decides to start using \(\frac16\) more than the amount of liquid called for in each recipe.
- How much of each ingredient will the baker use if the recipe calls for:
- \(1\frac12\) cups of milk?
- 3 tablespoons of oil?
- What is the constant of proportionality for the relationship between the amount of liquid called for in the recipe and the amount this baker uses?
Solution:
-
- \(1\frac34\) cups.
- \(3\frac12\) tablespoons.
- \(\frac76, 1.1\overline{6}\), or equivalent.
Percent Increase and Decrease
This week, your student is learning to describe increases and decreases as a percentage of the starting amount. For example, two different school clubs can gain the same number of students, but have different percent increases.
The computer club had 8 students. Then they gained 6 students.
This is a 75% increase, because \(6 \div 8 = 0.75\).
They now have 14 students, which is 175% of the starting amount.
\(1.75 \boldcdot 8 = 14\)
Here is a task to try with your student:
The photography club had 20 students. Then the number of students increased by 35%. How many students are in the photography club now?
Solution:
27 students. Possible strategies:
- The club gained 7 new students, because \(0.35 \boldcdot 20 = 7\). The club now has 27 students, because \(20 + 7 = 27\).
- The club now has 135% as many students as they started with, because \(100 + 35 = 135\). That means they have 27 students, because \(1.35 \boldcdot 20 = 27\).
Applying Percentages
This week, your student is learning about real-world situations that use percent increase and percent decrease, such as tax, interest, mark-up, and discounts.
For example, the price tag on a jacket says $24. The customer must also pay a sales tax equal to 7.5% of the price. What is the total cost of the jacket, including tax?
\(24 \boldcdot 1.075 = 25.80\)
The customer will pay 107.5% of the price listed on the tag, which is $25.80.
We can also find the percentage. For example, a backpack originally cost $22.50, but is on sale for $18.99. The discount is what percentage of the original price?
\(\begin{align} 22.50x &= 18.99 \\ x &= 18.99 \div 22.50 \\ x &= 0.844 \end{align}\)
The sale price is 84.4% of the original price. The discount is \(100 - 84.4\), or 15.6% of the original price.
Here is a task to try with your student:
A restaurant bill is $18.75. If you paid $22, what percentage tip did you leave for the server?
Solution:
\(17.\overline{3}\%\). Possible strategy: You paid \(117.\overline{3}\%\) of the bill, because \(22 \div 18.75 = 1.17\overline{3}\). You left a \(17.\overline{3}\%\) tip, because \(117.\overline{3} - 100 = 17.\overline{3}\).