Remember the GORP from the Mental Math for Addition and Subtraction Equations Concept?

Jason worked on the GORP and then handed it over for distribution. Some of the best foods to hike with are cheese, peanut butter, candy bars, fruit, beef jerky, dried fruit and energy bars. Jason was very excited about the GORP.

Travis and Henry two of the boys in Kelly’s group took a scoop, a scale, some gallon plastic bags and 24 pounds of GORP. Travis and Henry have to split up the GORP into bags so that each member of the group has the same amount. They have to gather GORP for the two leaders as well.

How much GORP should go into each bag?

Travis and Henry grab a piece of paper and a pencil and begin working this out.

**Wait a minute! An equation would be very helpful here. This Concept is all about writing and solving equations that have multiplication and division in them. You will learn all about equations and then revisit this problem at the end of the Concept.**

### Guidance

Previously we worked on single-variable addition and subtraction equations. To solve variable equations using multiplication and division we follow the same procedure. First we examine the problem and use mental math.

**Using mental math requires you to remember your multiplication tables. Remember that division is the opposite of multiplication, so when using small numbers mental math and your times tables will be your best strategy to balancing equations.**

\begin{align*}9p=72\end{align*}

**Here is a multiplication problem. Whenever you see a variable next to a number it means multiplication. Here we need to figure out, “What number times 9 is equal to 72?”**

**Using mental math, you can see that \begin{align*}p\end{align*} is equal to 8.**

**Now let’s check our answer.** We do this by substituting the value of the variable, 8, back into the original problem.

\begin{align*}9(8) &= 72\\ 72 &= 72\end{align*}

**One side of the equation is equal to the other side, so our equation balances! Our work is complete and accurate.**

**We can apply mental math when solving division problems too. Remember that a fraction bar means division, so when you see one you know that you are going to be dividing.**

\begin{align*}\frac{x}{3}=4\end{align*}

**Remember that when the variable is on top of the fraction bar that we are dividing the bottom number into this number. So we ask ourselves, “What number divided by three is equal to four?”**

**We use mental math to figure this out. Our missing value is 12.**

**Next, we can check our work.** We substitute 12 back into the equation for the variable \begin{align*}x\end{align*}.

\begin{align*}\frac{12}{3} &= 4\\ 4 &= 4\end{align*}

Practice solving these equations by using mental math.

#### Example A

\begin{align*}5y=20\end{align*}

**Solution: 4**

#### Example B

\begin{align*}6g = 42\end{align*}

**Solution: 7**

#### Example C

\begin{align*}\frac{x}{7}=2\end{align*}

**Solution: 14**

Now back to the GORP calculations.

Jason worked on the GORP and then handed it over for distribution. Some of the best foods to hike with are cheese, peanut butter, candy bars, fruit, beef jerky, dried fruit and energy bars. Jason was very excited about the GORP.

Travis and Henry two of the boys in Kelly’s group took a scoop, a scale, some gallon plastic bags and 24 pounds of GORP. Travis and Henry have to split up the GORP into bags so that each member of the group has the same amount. They have to gather GORP for the two leaders as well.

How much GORP should go into each bag?

Travis and Henry grab a piece of paper and a pencil and begin working this out.

First, let’s work on writing an equation.

There are 10 kids per group + 2 leaders = 12 total members of the group.

There is 24 lbs of GORP.

\begin{align*}12x=24\end{align*}

\begin{align*}x =\end{align*} **the number of pounds of GORP per person**

**Each person will have 2 pounds of GORP in their bag.**

### Vocabulary

- Algebraic Expression
- an expression that contains a combination of numbers, variables and operations. It does not have an equals sign.

- Equation
- a number sentence with two expressions divided by an equal sign. One quantity on one side of the equation equals the quantity on the other side of the equation.

- Variable Equation
- an equation where a variable is used to represent an unknown quantity.

### Guided Practice

Here is one for you to try on your own.

Alyssa sold $120 in raffle tickets. If each ticket costs $6, how many tickets did she sell? Write a variable equation and solve.

**Answer**

This problem will involve multiplication because the number of tickets times the cost of each ticket will equal the amount of money Alyssa made.

Let’s let \begin{align*}y\end{align*} represent the number of tickets Alyssa sold. The equation \begin{align*}6y = 120\end{align*}.

Using mental math, you should see the relationship between 6 and 12.

\begin{align*}6 \times 2 &= 12\\ 6 \times 20 &= 120\\ y &= 20\end{align*}

**Alyssa sold 20 raffle tickets. This is our answer.**

### Video Review

This is a James Sousa video on how to solve one-step equations by multiplication and division.

### Practice

Directions: 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*}