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# Mental Math for Multiplication/Division Equations

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Solve and Check Single-Variable Equations Using Mental Math and Substitution

Maria and her mother are shopping for a dress for Maria’s school dance. The dress Maria likes the most is 120. She asks her mom, “If I give you my cleaning money for the next six weeks, will you pay the other half?” Maria is paid the same amount of money each week to clean her neighbor’s house. How can you use the information given here to determine how much money Maria earns each week? In this concept, you will learn to solve and check single variable equations using mental math and substitution. ### Solving Equations Using Mental Math An equation is a statement of equality of two mathematical expressions. The quantity on one side of the equals sign must have the same value as the quantity on the other side of the equals sign. For example, 116=5\begin{align*}11-6=5\end{align*}, x+4=12\begin{align*}x+4 =12\end{align*}, 3x6=9\begin{align*}3x-6=9\end{align*}, and x25=4\begin{align*}\frac{x}{2}-5=4\end{align*} are all equations with a single variable. Remember, a variable is a letter that represents a quantity. When working with an equation involving a single variable, you are looking for the number that gives a true statement when you replace the variable by the number. This process is referred to as solving the equation. The number that gives the true statement is said to “satisfy” the equation and is called the solution or root of the equation. Let’s look at an example of solving an equation with a single variable. Solve the equation x+4=12\begin{align*}x+4=12\end{align*}. You can solve this equation using mental math. Ask yourself, “What number added to 4 gives 12 or what is the result of subtracting 4 from 12?” Either way, the solution is 8 because 8+4=12\begin{align*}8+4=12\end{align*} and 124=8\begin{align*}12-4=8\end{align*}. You have solved the equation and can express your solution as x=8\begin{align*}x=8\end{align*}. To check your answer, return to the equation you were given to solve and substitute x=8\begin{align*}x=8\end{align*} into the equation. x+48+4==1212\begin{align*}\begin{array}{rcl} x+4 & = & 12\\ 8+4 & = & 12 \end{array}\end{align*} Then, add the numbers on the left side of the equals sign. \begin{align*}\begin{array}{rcl} 8+4 & = & 12\\ 12 & = & 12 \end{array}\end{align*} The solution of \begin{align*}x=8\end{align*} gives a true statement. Let’s look at one more example. Solve the equation \begin{align*}3x=18\end{align*}. You can figure out this solution by using mental math. Ask yourself, “What number times 3 gives 18 or how many times 3 divides into 18?” Either way, the solution is 6 because \begin{align*}3 \times 6 =18\end{align*} and \begin{align*}18 \div 3 = 6\end{align*}. The solution is \begin{align*}x=6\end{align*}. To check your answer, return to the equation you were given to solve and substitute \begin{align*}x=6\end{align*} into the equation. \begin{align*}\begin{array}{rcl} 3x & = & 18\\ 3(6) & = & 18 \end{array}\end{align*} Now multiply the numbers on the left side of the equals sign. \begin{align*}18=18\end{align*} The solution of \begin{align*}x=6\end{align*} gives a true statement. ### Examples #### Example 1 Earlier, you were given a problem about Maria and her new dress. You want to know how much money Maria earns each week, if six weeks worth of pay is equal to half the cost of the120 dress.

First, divide \begin{align*}\120.00 \div 2\end{align*} to figure out how much Maria and her mom are each paying for the gown.

\begin{align*}\frac{120}{2}=60\end{align*}

Next, write an equation to show that Maria is paying 6 times some amount of money \begin{align*}(x)\end{align*} equal to \$60.

\begin{align*}6x=60\end{align*}

Then, ask yourself, “Six times what number equals 60?”

The solution is \begin{align*}x=10\end{align*} dollars.

#### Example 2

Solve the equation \begin{align*}3x-6=9\end{align*}.

The equation is asking you to find 3 times what number \begin{align*}(x)\end{align*} take away 6 equals nine.

You can use mental math to solve this equation.

First, ask yourself, “What number subtract 6 equals 9?”

Next, ask yourself, “Three times what number \begin{align*}(x)\end{align*} equals 15?”

\begin{align*}x=5\end{align*}

To check your answer, return to the equation you were given to solve and substitute \begin{align*}x=5\end{align*} into the equation.

\begin{align*}\begin{array}{rcl} 3x-6 & = & 9\\ 3(5)-6 & = & 9 \end{array}\end{align*}

First, multiply \begin{align*}3(5)=15\end{align*} on the left side of the equals sign.

\begin{align*}15-6=9\end{align*}

Next, subtract the numbers \begin{align*}15-6=9\end{align*} on the left side of the equals sign.

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

The root is \begin{align*}x=5\end{align*}.

#### Example 3

Solve and check the equation \begin{align*}\frac{x}{2}=12\end{align*} using mental math.

First, determine what the equation is asking you to find.

The equation is asking you to find what number \begin{align*}(x)\end{align*} divided by two equals 12.

The number is 24 because \begin{align*}24 \div 2=12\end{align*}.

The root is \begin{align*}x=24\end{align*}.

CHECK:

First, substitute \begin{align*}x=24\end{align*} into the equation.

\begin{align*}\frac{24}{2}=12\end{align*}

Then, divide 24 by 2 on the left side of the equals sign.

\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ 12}{2 \overline{ ) {24 }}}\\ && \ \underline{-2}\\ && \quad 04\\ && \ \ \ \underline{-4}\\ && \quad \ \ 0 \\ \\ && 12=12 \end{array}\end{align*}

#### Example 4

Determine whether \begin{align*}x=6\end{align*} is the solution to the following equation:

\begin{align*}3x-8=12\end{align*}

First, substitute \begin{align*}x=6\end{align*} into the given equation.

\begin{align*}3(6)-8=12\end{align*}

Next, multiply \begin{align*}3(6)=18\end{align*} to clear the parenthesis.

\begin{align*}18-8=12\end{align*}

Then, subtract the numbers on the left side of the equals sign. \begin{align*}18-8=10\end{align*}

\begin{align*}10=12\end{align*}

This is not a true statement.

The answer is NOT \begin{align*}x=6\end{align*}.

### Review

Solve each equation using mental math. Be sure to check each answer by substituting your solution back into the original problem. Then simplify to see if the equation expresses a true statement.

1. \begin{align*}x+4=22\end{align*}
2. \begin{align*}y+8=30\end{align*}
3. \begin{align*}x-19=40\end{align*}
4. \begin{align*}12-x=9\end{align*}
5. \begin{align*}4x=24\end{align*}
6. \begin{align*}6x=36\end{align*}
7. \begin{align*}9x=81\end{align*}
8. \begin{align*}\frac{y}{5}=2\end{align*}
9. \begin{align*}\frac{a}{8}=5\end{align*}
10. \begin{align*}\frac{12}{b}=6\end{align*}
11. \begin{align*}6x+3=27\end{align*}
12. \begin{align*}8y-2=54\end{align*}
13. \begin{align*}3b+12=30\end{align*}
14. \begin{align*}9y-7=65\end{align*}
15. \begin{align*}12a-5=31\end{align*}
16. \begin{align*}\frac{x}{2} + 4=8\end{align*}
17. \begin{align*}\frac{x}{4}+3=7\end{align*}
18. \begin{align*}\frac{10}{x} +9 =14\end{align*}
19. \begin{align*}5a-12=33\end{align*}
20. \begin{align*}7b-9=33\end{align*}

To see the Review answers, open this PDF file and look for section 1.11.

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### Vocabulary Language: English

TermDefinition
Algebraic Expression An expression that has numbers, operations and variables, but no equals sign.
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
Inverse Operation Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.
Variable A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.