# 1.12: Mental Math for Multiplication or Division Equations

Difficulty Level: At Grade Created by: CK-12
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Practice Single Variable Multiplication Equations

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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.

### Solving Multiplication and Division Problems Using Mental Math

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*}
• \begin{align*}12 = \frac{x}{3}\end{align*}
• \begin{align*}\frac{20}{x}+1 = 5\end{align*}

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*} equals 72”. The number next to the variable is called the coefficient of the variable. Here, 9 is the coefficient of \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*} that will make both sides of the equation equal.

“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*} must be 8.

The answer is \begin{align*}p=8\end{align*}.

You can check your solution by substituting that value for \begin{align*}p\end{align*} back into the original equation and verifying that it makes both sides equal.

\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*} is the same as \begin{align*}x\div 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*} must be 12.

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.

### Examples

#### Example 1

Earlier, you were given a problem about Olivia and her deck 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*} equal the number of cards in the deck.

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*} must be 50.

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.

#### Example 2

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*} equal the number of tickets that Alyssa sold.

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*} must be 20.

The answer is that Alyssa sold 20 tickets.

#### Example 3

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*} must be 4.

The answer is \begin{align*}y=4\end{align*}.

\begin{align*}\begin{array}{rcl} 5y & = & 20\\ 5(4) & = & 20\\ 20 & = & 20 \end{array}\end{align*}

#### Example 4

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*} must be 7.

The answer is \begin{align*}g=7\end{align*}.

\begin{align*}\begin{array}{rcl} 6g & = & 42\\ 6(7) & = & 42\\ 42 & = & 42 \end{array}\end{align*}

#### Example 5

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*} must be 14.

The answer is \begin{align*}x=14\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{x}{7} & = & 2 \\ \frac{14}{7} & = & 2\\ 2 & = & 2 \end{array}\end{align*}

### Review

Use mental math to solve each multiplication or division equation.

1. \begin{align*}5x = 25\end{align*}
2. \begin{align*}6x = 48\end{align*}
3. \begin{align*}2y = 18\end{align*}
4. \begin{align*}3y = 21\end{align*}
5. \begin{align*}4a = 16\end{align*}
6. \begin{align*}13b = 26\end{align*}
7. \begin{align*}15a = 30\end{align*}
8. \begin{align*}15x = 45\end{align*}
9. \begin{align*}\frac{x}{2}=3\end{align*}
10. \begin{align*}\frac{x}{4}=5\end{align*}
11. \begin{align*}\frac{x}{3}=11\end{align*}
12. \begin{align*}\frac{x}{5}=12\end{align*}
13. \begin{align*}\frac{x}{7}=8\end{align*}
14. \begin{align*}\frac{x}{8}=9\end{align*}
15. \begin{align*}\frac{x}{3}=12\end{align*}

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