Equation Word Problems 2

Question 1 of 5

The product of

$3$ and

$a$ is divided by

$4$ and is equal to twice the sum of

$a$ and

$10$ . Solve for

$a$ .

$a=$ (-16)

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

$a$

$(b+c)=$ $a$

$b+$ $a$

$c$

Keywords for Word Problems
Keyword	Meaning
Certain number	Unknown variable ( $x,y,z,$ etc)
is the same as, The answer is, is equal	$=$
sum, and, plus, more than	$+$
subtracted, less, minus	$-$
times, product, multiplied, twice	$times$
half of, divide by	$divide$

First, form an equation using the information given.

The product of $3$ and $a$	is divided by	$4$	and is equal to	twice	the sum of $a$ and $10$
$3timesa$	$divide$	$4$	$=$	$2times$	$a+10$

Simplify the equation:

$(3timesa)divide4$	$=$	$2times(a+10)$

$(3a)/4$	$=$	$2(a+10)$

To solve for $a$ , it needs to be alone on one side.

Start by removing

$1/4$ by multiplying both sides of the equation by

$4$ .

$\frac{3\color{#00880A}{a}}{4}$	$=$	$2($ $a$ $+10)$

$\frac{3\color{#00880A}{a}}{4}\color{#CC0000}{\times4}$	$=$	$2($ $a$ $+10)$ $times4$

$3$ $a$	$=$	$8($ $a$ $+10)$	$1/4times4$ cancels out

Next, expand the right side of the equation using the Distributive Property.

$3$ $a$	$=$	$8$ $($ $a$ $+10)$
$3$ $a$	$=$	$8$ $a$ $+$ $8$ $(10)$
$3$ $a$	$=$	$8$ $a$ $+80$

Now, move

$8a$ to the other side by subtracting

$8a$ from both sides of the equation.

$3$ $a$	$=$	$8$ $a$ $+80$
$3$ $a$ $-8a$	$=$	$8$ $a$ $+80$ $-8a$
$-5$ $a$	$=$	$80$	$8a-8a$ cancels out

Finally, remove

$-5$ by dividing both sides of the equation by

$-5$ .

$-5$ $a$	$=$	$80$
$-5$ $a$ $divide-5$	$=$	$80$ $divide-5$
$a$	$=$	$-16$	$-5divide-5$ cancels out

$a=-16$

Question 2 of 5

$3$ times a number plus

$7$ is the same as

$5$ times the number minus

$9$ . What is the number?

(8)

Question 3 of 5

A number divided by

$4$ plus

$6$ equals twice the number minus

$15$ . What is the number?

(12)

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

$a$

$(b+c)=$ $a$

$b+$ $a$

$c$

Keywords for Word Problems
Keyword	Meaning
Certain number	Unknown variable ( $x,y,z,$ etc)
is the same as, The answer is, is equal	$=$
sum, and, plus, more than	$+$
subtracted, less, minus	$-$
times, product, multiplied, twice	$times$
half of, divide by	$divide$

First, label the values and form an equation using the information given.

Number

$=x$

A number	divided by	$4$	plus	$6$	equals	twice	the number	minus	$15$
$x$	$divide$	$4$	$+$	$6$	$=$	$2times$	$x$	$-$	$15$

Simplify the equation:

$(xdivide4)+6$	$=$	$(2timesx)-15$

$x/4 +6$	$=$	$2x-15$

To solve for $x$ , it needs to be alone on one side.

Start by removing

$1/4$ by multiplying both sides of the equation by

$4$ .

$\frac{\color{#00880A}{x}}{4}+6$	$=$	$2$ $x$ $-15$

$\left(\frac{\color{#00880A}{x}}{4}+6\right)\color{#CC0000}{\times4}$	$=$	$(2$ $x$ $-15)$ $times4$

$x$ $+24$	$=$	$4(2$ $x$ $-15)$	$1/4times4$ cancels out

Next, expand the right side of the equation using the Distributive Property.

$x$ $+24$	$=$	$4$ $(2$ $x$ $-15)$
$x$ $+24$	$=$	$4$ $(2$ $x$ $)+$ $4$ $(-15)$
$x$ $+24$	$=$	$8$ $x$ $-60$

Now, move

$x$ to the other side by subtracting

$x$ from both sides of the equation.

$x$ $+24$	$=$	$8$ $x$ $-60$
$x$ $+24$ $-x$	$=$	$8$ $x$ $-60$ $-x$
$24$	$=$	$7$ $x$ $-60$	$x-x$ cancels out

Then, move

$60$ to the other side by adding

$60$ to both sides of the equation.

$24$	$=$	$7$ $x$ $-60$
$24$ $+60$	$=$	$7$ $x$ $-60$ $+60$
$84$	$=$	$7$ $x$	$-60+60$ cancels out

Finally, remove

$7$ by dividing both sides of the equation by

$7$ .

$84$	$=$	$7$ $x$
$84$ $divide7$	$=$	$7$ $x$ $divide7$
$12$	$=$	$x$	$7divide7$ cancels out
$x$	$=$	$12$

The number is

$12$ .

$12$

Question 4 of 5

The sum of a number and

$6$ when multiplied by

$4$ equals

$3$ times the sum of

$9$ and that number. What is the number?

(3)

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

$a$

$(b+c)=$ $a$

$b+$ $a$

$c$

Keywords for Word Problems
Keyword	Meaning
Certain number	Unknown variable ( $x,y,z,$ etc)
is the same as, The answer is, is equal	$=$
sum, and, plus, more than	$+$
subtracted, less, minus	$-$
times, product, multiplied, twice	$times$
half of, divide by	$divide$

First, label the values and form an equation using the information given.

Number

$=x$

The sum of a number and $6$	when multiplied by	$4$	equals	$3$ times	the sum of $9$ and that number
$x+6$	$times$	$4$	$=$	$3times$	$9+x$

Simplify the equation:

$(x+6)times4$	$=$	$3times(9+x)$
$4(x+6)$	$=$	$3(9+x)$

To solve for $x$ , it needs to be alone on one side.

Start by expanding both sides of the equation by using the Distributive Property.

$4$ $($ $x$ $+6)$	$=$	$3$ $(9+$ $x$ $)$
$4$ $x$ $+$ $4$ $(6)$	$=$	$3$ $(9)+$ $3$ $x$
$4$ $x$ $+24$	$=$	$27+3$ $x$

Now, move

$3x$ to the other side by subtracting

$3x$ from both sides of the equation.

$4$ $x$ $+24$	$=$	$27+3$ $x$
$4$ $x$ $+24$ $-3x$	$=$	$27+3$ $x$ $-3x$
$x$ $+24$	$=$	$27$	$3x-3x$ cancels out

Finally, move

$24$ to the other side by subtracting

$24$ from both sides of the equation.

$x$ $+24$	$=$	$27$
$x$ $+24$ $-24$	$=$	$27$ $-24$
$x$	$=$	$3$	$24-24$ cancels out

The number is

$3$ .

Check our work

To confirm our answer, substitute

$x=3$ to the original equation.

$4(x+6)$	$=$	$3(9+x)$
$4(3+6)$	$=$	$3(9+3)$	Substitute $x=3$
$4(9)$	$=$	$3(12)$
$36$	$=$	$36$

Since the equation is true, the answer is correct.

$3$

Question 5 of 5

$24$ plus

$3$ times a number equals

$4$ divided into the sum of 7 times the number plus

$1$ . What is the number?

(-19)

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Inverse Operations

When moving a term to the other side of an equation, the operation is inversed.

Distributive Property

$a$

$(b+c)=$ $a$

$b+$ $a$

$c$

Keywords for Word Problems
Keyword	Meaning
Certain number	Unknown variable ( $x,y,z,$ etc)
is the same as, The answer is, is equal	$=$
sum, and, plus, more than	$+$
subtracted, less, minus	$-$
times, product, multiplied, twice	$times$
half of, divide by	$divide$

First, label the values and form an equation using the information given.

Number

$=n$

$24$	plus	$3$	times	a number	equals	$4$ divided into the sum of 7 times the number plus $1$
$24$	$+$	$3$	$times$	$n$	$=$	$[(7timesn)+1]divide4$

Simplify the equation:

$24+(3timesn)$	$=$	$[(7timesn)+1]divide4$

$24+3n$	$=$	$(7n+1)/4$

To solve for $n$ , it needs to be alone on one side.

Start by removing

$1/4$ by multiplying both sides of the equation by

$4$ .

$24+3$ $n$	$=$	$\frac{7\color{#00880A}{n}+1}{4}$

$(24+3$ $n$ $)$ $times4$	$=$	$\frac{7\color{#00880A}{n}+1}{4}\color{#CC0000}{\times4}$

$4(24+3$ $n$ $)$	$=$	$7$ $n$ $+1$	$1/4times4$ cancels out

Next, expand the left side of the equation using the Distributive Property.

$4$ $(24+3$ $n$ $)$	$=$	$7$ $n$ $+1$
$4$ $(24)+$ $4$ $(3$ $n$ $)$	$=$	$7$ $n$ $+1$
$96+12$ $n$	$=$	$7$ $n$ $+1$

Now, move

$7n$ to the other side by subtracting

$7n$ from both sides of the equation.

$96+12$ $n$	$=$	$7$ $n$ $+1$
$96+12$ $n$ $-7n$	$=$	$7$ $n$ $+1$ $-7n$
$96+5$ $n$	$=$	$1$	$7n-7n$ cancels out

Then, move

$96$ to the other side by subtracting

$96$ from both sides of the equation.

$96+5$ $n$	$=$	$1$
$96+5$ $n$ $-96$	$=$	$1$ $-96$
$5$ $n$	$=$	$-95$	$96-96$ cancels out

Finally, remove

$5$ by dividing both sides of the equation by

$5$ .

$5$ $n$	$=$	$-95$
$5$ $n$ $divide5$	$=$	$-95$ $divide5$
$n$	$=$	$-19$	$5divide5$ cancels out

The number is

$-19$ .

Check our work

To confirm our answer, substitute

$n=-19$ to the original equation.

$24+3n$	$=$	$(7n+1)/4$

$24+3(-19)$	$=$	$(7(-19)+1)/4$	Substitute $n=-19$

$24-57$	$=$	$(-133+1)/4$

$-33$	$=$	$(-132)/4$

$-33$	$=$	$-33$

Since the equation is true, the answer is correct.

$-19$

$(3timesn)+7$	$=$	$(5timesn)-9$
$3n+7$	$=$	$5n-9$

$3$ $n$ $+7$	$=$	$5$ $n$ $-9$
$3$ $n$ $+7$ $-3n$	$=$	$5$ $n$ $-9$ $-3n$
$7$	$=$	$2$ $n$ $-9$	$3n-3n$ cancels out

$7$	$=$	$2$ $n$ $-9$
$7$ $+9$	$=$	$2$ $n$ $-9$ $+9$
$16$	$=$	$2$ $n$	$-9+9$ cancels out

$3n+7$	$=$	$5n-9$
$3(8)+7$	$=$	$5(8)-9$	Substitute $n=8$
$24+7$	$=$	$40-9$
$31$	$=$	$31$

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Quiz summary

Information

1. Question

Hint

Excellent!

Inverse Operations

Distributive Property

2. Question

Hint

Great Work!

Inverse Operations

3. Question

Hint

Well Done!

Inverse Operations

Distributive Property

4. Question

Hint

Fantastic!

Inverse Operations

Distributive Property

5. Question

Hint

Good Job!

Inverse Operations

Distributive Property

Quizzes

$16$	$=$	$2$ $n$
$16$ $divide2$	$=$	$2$ $n$ $divide2$
$8$	$=$	$n$	$2divide2$ cancels out
$n$	$=$	$8$