# Multiplying Mixed Numbers: Word Problems

## Multiplying Mixed Numbers – Word Problems

Understanding how to work with mixed numbers is essential in our daily lives. From cooking recipes to measuring materials for a project, we often encounter situations where mixed numbers come into play. Gaining comfort in multiplying these numbers helps in effectively solving real-world problems.

## Mixed Numbers and Improper Fractions

It's really helpful to know about improper fractions and mixed numbers when you're solving math problems, so let's make sure we understand what they are!

Improper Fraction: A fraction where the numerator (top number) is greater than or equal to the denominator (bottom number), such as $\frac{7}{2}$.

Mixed Number: A number consisting of a whole number and a fraction, like $3 \frac{1}{2}$.

## Conversion between Mixed Numbers and Improper Fractions

Process Description Example
Mixed to Improper Conversion Multiply the whole number by the denominator, add the numerator, then place over the denominator. $2 \frac{1}{2}$ → $\frac{2 \times 2 + 1}{2}$ = $\frac{5}{2}$
Improper to Mixed Conversion Divide the numerator by the denominator. The quotient is the whole number, the remainder over the denominator is the fraction. $\frac{7}{4}$ → $1 \frac{3}{4}$

Convert the mixed number $3 \frac{1}{4}$ to an improper fraction.

• Multiply the whole number part by the denominator and add the numerator.
• $3 \times 4 + 1 = 12 + 1 = 13$.
• The mixed number $3 \frac{1}{4}$ is equal to the improper fraction $\frac{13}{4}$.

Convert the improper fraction $\frac{11}{3}$ to a mixed number.

• Divide the numerator by the denominator.
• $11 \div 3 = 3$ remainder $2$.
• The improper fraction $\frac{11}{3}$ is equal to the mixed number $3 \frac{2}{3}$.
Convert the mixed number $2 \frac{3}{5}$ to an improper fraction.
Convert the improper fraction $\frac{15}{4}$ to a mixed number.
Convert the mixed number $4 \frac{1}{6}$ to an improper fraction.
Convert the improper fraction $\frac{22}{7}$ to a mixed number.

## Multiplying Mixed Numbers: Word Problems – Step-by-Step Process

Multiplying a fraction by a whole number and dealing with two mixed numbers are common in real-world problems. It's important to understand how these calculations work in practical situations. Let’s practice some examples!

A recipe needs $1 \frac{2}{3}$ cups of sugar for a batch of cookies, and you want to make $2 \frac{1}{2}$ batches.

• Convert Mixed to Improper: Convert $1 \frac{2}{3}$ to $\frac{5}{3}$ and $2 \frac{1}{2}$ to $\frac{5}{2}$.
• Multiply: Multiply $\frac{5}{3}$ by $\frac{5}{2}$.
• Calculate & Convert Back: $\frac{5}{3} \times \frac{5}{2} = \frac{25}{6}$. Convert $\frac{25}{6}$ to mixed number: $4 \frac{1}{6}$. You need $4 \frac{1}{6}$ cups of sugar.

If a garden requires $2 \frac{1}{4}$ cubic meters of soil for each section, and you are landscaping $3$ sections, how much soil is needed in total?

• Convert Mixed to Improper: Convert $2 \frac{1}{4}$ to an improper fraction: $\frac{2 \times 4 + 1}{4} = \frac{9}{4}$.
• Multiply by 3: $\frac{9}{4} \times 3$.
• Calculate & Convert Back: $\frac{27}{4} = 6 \frac{3}{4}$. You need $6 \frac{3}{4}$ cubic meters of soil.

Each shelf needs $2 \frac{1}{2}$ feet of wood, and you are building $4$ shelves. How much wood in total**?

• Convert Mixed to Improper: Convert $2 \frac{1}{2}$ to $\frac{5}{2}$.
• Multiply: Multiply by 4: $\frac{5}{2} \times 4$.
• Calculate & Convert Back: $\frac{20}{2} = 10$. You need $10$ feet of wood.

## Multiplying Mixed Numbers: Word Problems – Practice

Your garden has an area of $4 \frac{1}{5}$ square meters, and you want to add compost at a rate of $2 \frac{2}{3}$ kilograms per square meter. How much compost will you need?
Sarah is making cookies and needs to use $2 \frac{1}{3}$ cups of sugar for each batch. How much sugar will she need for 3 batches?
Each shelf in a bookcase can hold $5 \frac{7}{8}$ books. If there are 4 shelves, how many books can the bookcase hold in total?
A recipe for a cake requires $1 \frac{1}{4}$ cups of flour. If you decide to make half of the recipe, how much flour do you need?
John walks $3 \frac{2}{3}$ miles every day. How far will he walk in 5 days?
If a bag holds $7 \frac{1}{2}$ pounds of potatoes, how much do 6 bags weigh?

## Multiplying Mixed Numbers: Word Problems – Summary

Key Learnings from this Text:

• Multiplying mixed numbers is a practical skill for real-world applications.
• Conversion between mixed and improper fractions is key in multiplying mixed numbers.
• Multiplying mixed numbers involves converting to improper fractions, multiplying, and converting back.
• Understanding these steps helps in efficiently solving problems involving measurements, recipes, etc.

## Multiplying Mixed Numbers: Word Problems – Frequently Asked Questions

What is a mixed number in math?
How do you multiply two mixed numbers?
Why do we need to convert mixed numbers to improper fractions for multiplication?
Can you give an example of a real-world problem involving multiplying mixed numbers?
How do you convert an improper fraction to a mixed number?
What is an improper fraction?
Is it necessary to convert back to a mixed number after multiplying?
Can you multiply a mixed number by a whole number?
What does the result of multiplying mixed numbers represent in real life?
How do you simplify the result of multiplying mixed numbers?

## Multiplying Mixed Numbers: Word Problems exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Multiplying Mixed Numbers: Word Problems.
• ### A contractor needs $3 \frac{3}{4}$​ meters of wood to build one section of a fence.

Hints

Convert the mixed number $3 \frac{3}{4}$​​​ into an improper fraction by multiplying the whole number by the denominator of the fraction part and then adding the numerator.

Multiply the resulting improper fraction by the number of sections to find the total amount of wood needed.

Solution

First, we convert $3 \frac{3}{4}$​​ to an improper fraction.

Then, convert $3 \frac{3}{4}$​​ to an improper fraction:

$3 \times 4 + 3 = 15 \rightarrow \frac{15}{4}$​

Next, multiply the improper fraction by the number of sections:

$\frac{15}{4} \times 4 = \frac{60}{4}​ = 15$

Finally, the correct answer is $15$ meters.

• ### Julia is hosting a pizza party and each guest eats $2 \frac{1}{2}$ slices of pizza.

Hints

Convert $2 \frac{1}{2}$​ to an improper fraction:

$2 \times 2 + 1 = 5 \rightarrow \frac{5}{2}$​

Multiply the improper fraction by the number of guests:

$\frac{5}{2}​ \times 5 = \frac{25}{2}$​

Convert $\frac{25}{2}$​​ back to a mixed number.

Solution

They eat $12 \frac{1}{2}$ slices of pizza.

• ### A recipe calls for $1 \frac{1}{3}$ cups of sugar to make a batch of cookies.

Hints

Convert $1 \frac{1}{3}$ to an improper fraction:

$1 \times 3 + 1 = 4 \rightarrow \frac{4}{3}$

Multiply the improper fraction by the number of batches:

$\frac{4}{3} \times 3$

Solution

The total amount of sugar needed to make 3 batches of cookies is 4 cups.

1. Convert $1 \frac{1}{3}$ to an improper fraction: $1 \times 3 + 1 = 4 \rightarrow \frac{4}{3}$

2. Multiply the improper fraction by the number of batches: $\frac{4}{3} \times 3 = \frac{12}{3} = 4$

• ### In an art class, each student uses $1 \frac{3}{5}$ tubes of paint for their project.

Hints

Convert the mixed number $1 \frac{3}{5}$​ into an improper fraction by multiplying the whole number part by the denominator of the fractional part and then adding the numerator.

Multiply the resulting improper fraction by the number of students to find out the total number of tubes needed for the class, and then convert the answer to a mixed number if necessary.

Solution

$12\mathbf{\frac{4}{5}}$ tubes of paint are used by all the students.

1. Convert $1 \frac{3}{5}$ to an improper fraction: $1 \times 5 + 3 = 8 \rightarrow \frac{8}{5}$

2. $\frac{8}{5} \times 8 = \frac{64}{5} = 12 \frac{4}{5}$

• ### During a relay race, each runner covers $2 \frac{3}{4}$​ kilometers.

Hints

Convert the mixed number $2 \frac{3}{4}$​ ​into an improper fraction by multiplying the whole number by the denominator of the fraction part and then adding the numerator.

After converting, multiply the resulting improper fraction by the number of runners to find the total distance covered by the team.

Solution

Convert $2 \frac{3}{4}$​ to an improper fraction and then multiply the fraction by the number of runners.

The total distance covered by the team is $\bf{11}$ kilometers.

1.Convert $2 \frac{3}{4}$​ to an improper fraction: $2 \times 4 + 3 = 11 \rightarrow \frac{11}{4}$​

2. Multiply the improper fraction by the number of runners: $\frac{11}{4}​ \times 4 = \frac{44}{4} = 11$

• ### How many liters of water are needed to fill the pool?

Hints

Convert the mixed number $5 \frac{3}{8}$​ into an improper fraction by multiplying the whole number part by the denominator of the fractional part and then adding the numerator.

Multiply the resulting improper fraction by the number of buckets to find out the total amount of water needed to fill the pool, and then convert the answer to a mixed number if necessary.

Solution

The total amount of water needed to fill the pool is $32 \frac{1}{4}$ liters.

1. Convert $5 \frac{3}{8}$​ to an improper fraction: $5 \times 8 + 3 = 43 \rightarrow \frac{43}{8}$

2. Multiply the improper fraction by the number of buckets: $\frac{43}{8} \times 6 = \frac{258}{8} = 32 \frac{2}{8} = 32 \frac{1}{4}$

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