Lesson 12
Solving Problems Involving Fractions
Let’s add, subtract, multiply, and divide fractions.
12.1: Operations with Fractions
Without calculating, order the expressions according to their values from least to greatest. Be prepared to explain your reasoning.
\(\frac34 + \frac23\)
\(\frac34  \frac23\)
\(\frac34 \boldcdot \frac23\)
\(\frac34 \div \frac23\)
12.2: Situations with $\frac34$ and $\frac12$
Here are four situations that involve \(\frac34\) and \(\frac12\).
 Before calculating, decide if each answer is greater than 1 or less than 1.
 Write a multiplication equation or division equation for the situation.
 Answer the question. Show your reasoning. Draw a tape diagram, if needed.
 There was \(\frac34\) liter of water in Andre’s water bottle. Andre drank \(\frac12\) of the water. How many liters of water did he drink?
 The distance from Han’s house to his school is \(\frac34\) kilometers. Han walked \(\frac12\) kilometers. What fraction of the distance from his house to the school did Han walk?
 Priya’s goal was to collect \(\frac12\) kilograms of trash. She collected \(\frac34\) kilograms of trash. How many times her goal was the amount of trash she collected?
 Mai’s class volunteered to clean a park with an area of \(\frac 12\) square mile. Before they took a lunch break, the class had cleaned \(\frac 34\) of the park. How many square miles had they cleaned before lunch?
12.3: Pairs of Problems

Work with a partner to write equations for the following questions. One person works on the questions labeled A1, B1, . . . , E1 and the other person works on those labeled A2, B2, . . . , E2.
A1. Lin’s bottle holds \(3 \frac 14\) cups of water. She drank 1 cup of water. What fraction of the water in the bottle did she drink?
A2. Lin’s bottle holds \(3 \frac 14\) cups of water. After she drank some, there were \(1 \frac 12\) cups of water in the bottle. How many cups did she drink?
B1. Plant A is \( \frac{16}{3}\) feet tall. This is \(\frac 45\) as tall as Plant B. How tall is Plant B?
B2. Plant A is \(\frac{16}{3}\) feet tall. Plant C is \(\frac 45\) as tall as Plant A. How tall is Plant C?
C1. \(\frac 89\) kilogram of berries is put into a container that already has \( \frac 73\) kilogram of berries. How many kilograms are in the container?
C2. A container with \(\frac 89\) kilogram of berries is \(\frac 23\) full. How many kilograms can the container hold?
D1. The area of a rectangle is \(14\frac12\) sq cm and one side is \(4 \frac 12\) cm. How long is the other side?
D2. The side lengths of a rectangle are \(4 \frac 12\) cm and \(2 \frac 25\) cm. What is the area of the rectangle?
E1. A stack of magazines is \(4 \frac 25\) inches high. The stack needs to fit into a box that is \(2 \frac 18\) inches high. How many inches too high is the stack?
E2. A stack of magazines is \(4\frac 25\) inches high. Each magazine is \(\frac 25\)inch thick. How many magazines are in the stack?

Trade papers with your partner, and check your partner’s equations. If you disagree, work to reach an agreement.

Your teacher will assign 2 or 3 questions for you to answer. For each question:
 Estimate the answer before calculating it.
 Find the answer, and show your reasoning.
12.4: Baking Cookies
Mai, Kiran, and Clare are baking cookies together. They need \(\frac 34\) cup of flour and \(\frac 13\) cup of butter to make a batch of cookies. They each brought the ingredients they had at home.

Mai brought 2 cups of flour and \(\frac 14\) cup of butter.

Kiran brought 1 cup of flour and \(\frac 12\) cup of butter.

Clare brought \(1\frac 14\) cups of flour and \(\frac34\) cup of butter.
If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of cookies can they make? Explain your reasoning.
Summary
We can add, subtract, multiply, and divide both whole numbers and fractions. Here is a summary of how we add, subtract, multiply, and divide fractions.
 To add or subtract fractions, we often look for a common denominator so the pieces involved are the same size. This makes it easy to add or subtract the pieces.
\(\displaystyle \frac32  \frac45 = \frac{15}{10}  \frac{8}{10}\)
 To multiply fractions, we often multiply the numerators and the denominators.
\(\displaystyle \frac38 \boldcdot \frac59 = \frac{3 \boldcdot 5}{8 \boldcdot 9}\)
 To divide a number by a fraction \(\frac ab\), we can multiply the number by \(\frac ba\), which is the reciprocal of \(\frac ab\).
\(\displaystyle \frac47 \div \frac53 = \frac47 \boldcdot \frac35\)