1.12: Mental Math for Multiplication or Division Equations
Let’s Think About It
Olivia loves to play cards with her friends. She knows that normally a deck of cards has 52 cards. Unfortunately, her baby brother was playing with her deck of cards and she thinks some of the cards have gone missing. When she deals out all of the cards to herself and her 4 friends, each person ends up with 10 cards. How can Olivia figure out how many cards are in her deck? Is she missing any cards?
In this concept, you will learn how to solve single variable multiplication and division equations using mental math.
Guidance
An equation is a mathematical sentence that says that two mathematical expressions are equal. A variable equation is an equation that contains a variable.
Here are some examples of variable equations.

\begin{align*}15t = 45\end{align*}
15t=45

\begin{align*}12 = \frac{x}{3}\end{align*}
12=x3

\begin{align*}\frac{20}{x}+1 = 5\end{align*}
20x+1=5
When you solve an equation you are looking for the value of the variable that makes both sides of the equation equal. This value of the variable that makes both sides of the equation equal is called the solution.
Some variable equations are simple enough that you are able to determine the solution to the equation in your head using mental math.
Here is an example that involves multiplication.
Use mental math to solve the following equation.
\begin{align*}9p=72\end{align*}
First, remember that whenever you see a number right next to a variable it means the number is being multiplied by the variable. This equation says “9 times \begin{align*}p\end{align*}
Now, turn the equation into a question that you can ask yourself. Remember you are trying to figure out the value of \begin{align*}p\end{align*}
“9 times what number is equal to 72?”
Next, answer your question. You know that 9 times 8 is equal to 72. This means \begin{align*}p\end{align*}
The answer is \begin{align*}p=8\end{align*}
You can check your solution by substituting that value for \begin{align*}p\end{align*}
\begin{align*}\begin{array}{rcl} 9p & = & 72\\ 9(8) & = & 72\\ 72 & = & 72 \end{array}\end{align*}
Both sides of the equation are equal so your answer is correct!
Here is an example that involves division.
Use mental math to solve the following equation.
\begin{align*}\frac{x}{3}=4\end{align*}
First, remember that a fraction bar means division. \begin{align*}\frac{x}{3}\end{align*}
Now, turn the equation into a question that you can ask yourself.
“What number divided by 3 is equal to 4?”
Next, answer your question. You know that 12 divided by 3 is equal to 4. This means \begin{align*}x\end{align*}
The answer is \begin{align*}x=12\end{align*}
Make sure to check your solution.
\begin{align*}\begin{array}{rcl} \frac{x}{3} & = & 4 \\ \frac{12}{3} & = & 4\\ 4 & = & 4 \end{array}\end{align*}
Because both sides of the equation are equal, you can be confident that your answer is correct.
Guided Practice
Alyssa sold $120 in raffle tickets. If each ticket costs $6, how many tickets did she sell? Write a variable equation and solve.
First, define your variable. You don't know how many tickets Alyssa sold so this unknown quantity will be your variable.
Let \begin{align*}x\end{align*}
Next, you know that Alyssa sold each ticket for $6 and made $120 total. This scenario involves multiplication because the number of tickets times the cost of each ticket will equal the amount of money Alyssa made.
\begin{align*}6x=120\end{align*}
Now, turn the equation into a question that you can ask yourself.
“6 times what number is equal to 120?”
Then, answer your question. You know that 6 times 2 is equal to 12, so 6 times 20 is equal to 120. This means \begin{align*}x\end{align*}
The answer is that Alyssa sold 20 tickets.
Examples
Example 1
Use mental math to solve the following equation.
\begin{align*}5y=20\end{align*}
First, turn the equation into a question that you can ask yourself.
“5 times what number is equal to 20?”
Next, answer your question. You know that 5 times 4 is equal to 20. This means \begin{align*}y\end{align*}
The answer is \begin{align*}y=4\end{align*}
Now, check your solution.
\begin{align*}\begin{array}{rcl} 5y & = & 20\\ 5(4) & = & 20\\ 20 & = & 20 \end{array}\end{align*}
Your answer is correct!
Example 2
Use mental math to solve the following equation.
\begin{align*}6g=42\end{align*}
First, turn the equation into a question that you can ask yourself.
“6 times what number is equal to 42?”
Next, answer your question. You know that 6 times 7 is equal to 42. This means \begin{align*}g\end{align*}
The answer is \begin{align*}g=7\end{align*}
Now, check your solution.
\begin{align*}\begin{array}{rcl} 6g & = & 42\\ 6(7) & = & 42\\ 42 & = & 42 \end{array}\end{align*}
Your answer is correct!
Example 3
Use mental math to solve the following equation.
\begin{align*}\frac{x}{7}=2\end{align*}
First, turn the equation into a question that you can ask yourself.
“What number divided by 7 is equal to 2?”
Next, answer your question. You know that 14 divided by 7 is equal to 2. This means \begin{align*}x\end{align*}
The answer is \begin{align*}x=14\end{align*}
Now, check your solution.
\begin{align*}\begin{array}{rcl} \frac{x}{7} & = & 2 \\ \frac{14}{7} & = & 2\\ 2 & = & 2 \end{array}\end{align*}
Your answer is correct!
Follow Up
Remember Olivia and her deck of cards that might be missing some cards? When she deals out all the cards to herself and 4 of her friends, each person ends up with 10 cards. She wants to know how many cards are in her deck.
First, define your variable. You don't know how many cards are in the deck so this unknown quantity will be your variable.
Let \begin{align*}x\end{align*}
Next, you know that when all the cards in the deck are divided evenly between 5 people, each person ends up with 10 cards. This scenario involves division because the total number of cards divided by the 5 people will equal the 10 cards each person got.
\begin{align*}\frac{x}{5}=10\end{align*}
Now, turn the equation into a question that you can ask yourself.
“What number divided by 5 is equal to 10?”
Then, answer your question. You know that 50 divided by 5 is equal to 10. This means \begin{align*}x\end{align*}
The answer is that Olivia's deck only has 50 cards. Since a deck of cards should have 52 cards, Olivia is missing 2 cards.
Explore More
Use mental math to solve each multiplication or division equation.

\begin{align*}5x = 25\end{align*}
5x=25 
\begin{align*}6x = 48\end{align*}
6x=48 
\begin{align*}2y = 18\end{align*}
2y=18 
\begin{align*}3y = 21\end{align*}
3y=21 
\begin{align*}4a = 16\end{align*}
4a=16 
\begin{align*}13b = 26\end{align*}
13b=26 
\begin{align*}15a = 30\end{align*}
15a=30 
\begin{align*}15x = 45\end{align*}
15x=45 
\begin{align*}\frac{x}{2}=3\end{align*}
x2=3 
\begin{align*}\frac{x}{4}=5\end{align*}
x4=5 
\begin{align*}\frac{x}{3}=11\end{align*}
x3=11 
\begin{align*}\frac{x}{5}=12\end{align*}
x5=12 
\begin{align*}\frac{x}{7}=8\end{align*}
x7=8  \begin{align*}\frac{x}{8}=9\end{align*}
 \begin{align*}\frac{x}{3}=12\end{align*}
Image Attributions
 [1]^ Credit: jesiehart; Source: https://www.flickr.com/photos/jessiehart/880972903/; License: CC BYNC 3.0
 [2]^ Credit: Abulic Monkey; Source: https://www.flickr.com/photos/abulic_monkey/537113962/; License: CC BYNC 3.0
Description
Learning Objectives
In this concept, you will learn how to solve single variable multiplication and division equations using mental math.
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Date Created:
Aug 10, 2015Last Modified:
Aug 26, 2015Vocabulary
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