Back to: Algebra 1

Most of the time when someone says “word problems” there is automatic panic. But word problems do not have to be a cause for concern. Set up a system and follow it.

Follow this guide.

**Read the problem carefully and figure out what it is asking you to find.****Assign a variable to the quantity you are trying to find.****Write down what the variable represents.****Re-read the problem and write an equation for the quantities given in the problem.****Solve the equation.****Answer the question in the problem.****Check your solution.**

**Example:** When 12 is added to five times a number, the result is 52. Find the number.

**Step 1: ****What are we trying to find?**

**A number****.**

**Step 2: ****Assign a variable for the number.**

**Let us call it x.**

**Step 3: ****Write down what the **variable** represents****.**

**Let ***x ***= ****a number**

**Step 4: ****Write an equation****.**

We are told 12 is added to 5 times a number. Since n represents the number, five times thenumber would be **5 x.** If 12 is added to that, we get

**5**We know that answer is 52 sonowwe have an equation, 5

*x*+ 12.*x*+ 12 = 52.

**Step 5: ****Solve the equation.**

**5 x +12 = 52**

**– 12 -12**

**5 x = 40 **

**—————**

*x* = 8

**Step 6**: Answer the question in the problem

The problem asks us to find a number. We decided that *x* would be the number, so we have *x* =**8**. The number we are looking for is **8.**

**Step 7: ****Check the answer.**

** 5(8) + 12 = 40 + 12 = 52 **

The answer makes sense.

**Example**: Three times the sum of a number and 5 is equal to five times the difference obtained

when 3 subtracted from the number.

**Step 1: ****What are we trying to find?**

**A number****.**

**Step 2: ****Assign a variable for the number.**

**Let us call it n.**

**Step 3: ****Write down what the **variable** represents****.**

**Let n = **

**a number**

**Step 4: ****Write an equation****.**

First, we are told three times the sum of a number and 5. Since *n* represents the number, three times thesum of the number and** 5** would be **3( n + 5).** If 3 is subtracted from that, we get

**3(**

*n*+ 5) – 3**.**We know that is equal to 5 times the number,

*n*, sonowwe have an equation,

3(*n* – 5) – 3 = 5*n*.

**Step 5: ****Solve the equation.**

**3( n + 5) – 3 = 5n**

3*n* + 15 – 3 = 5*n*

3*n* + 12 = 5*n*

12 = 2*n*

*n* = 6

**Step 6**: Answer the question in the problem

*n* = 6

**Step 7: ****Check the answer.**

3(6 + 5) – 3 = 5(6)

3(11) – 3 = 30

33 – 3 = 30

30 = 30.

**Example**: The length of a rectangular patio is 25 feet and the perimeter is 80 feet. Find the width.

**Step 1: ****What are we trying to find?**

**The width of a rectangle****.**

**Step 2: ****Assign a variable for the number.**

**Let us call it w.**

**Step 3: ****Write down what the variable represents****.**

**Let ***w ***= ****The width of the rectangular patio****.**

**Step 4: ****Write an equation****.**

The length is given as 25 feet and the perimeter 80 feet. The formula for the perimeter of a rectangle is ** 2l + 2w or p = 2l + 2w**. the equation is, 80 = 50 + 2w.

**Step 5: ****Solve the equation.**

80 = 50 + 2w

– 50 = – 50

—————

30 = 2w

15 = w

**Step 6**: **Answer the question in the problem**

The width of the rectangular patio is 15 feet.

**Step 7: ****Check the answer.**

2l + 2w = p

2(25) + 2(15) = 80

50 + 30 = 80.

**Practice Problems 4**

- Allan earns $300 less than Ben earns. Together they earn $2700. How much does Ben earn?
- The sum of 3 times a number and 15 is equal to 135. Find the number.
- The sum of the smallest and the largest of three consecutive numbers is 42. Find the three numbers.
- Lena received 3,500 likes on her Instagram page in the first month. In the third month she received three times as much as she did in the second month. Altogether, she received 23,500 over the three-month period. How many likes did she get in the second month?
- Jan is four times as old as her daughter Pat. In 9 years, she will be five times as old as Pat is now. How old is Pat now?
- I have a length of rope that is 76 inches long. It must be cut into two pieces so that one piece is 16 inches longer than the other. Write an equation and solve it to find the two lengths.
- The width of a rectangle is 18 inches and the perimeter is 86 inches. Find the length of the rectangle.
- The perimeter of a triangular lot is 90 meters. The first side is 20 meters, and the third side is twice the length of the second side. Find the length of the third side.
- If three times a number is decreased by 4, the result would be equal to twice the number increased by 4. Find the number.
- The larger of two numbers is 2 less than four times the smaller. The difference between nine times the smaller and two times the larger is 7. Find the two numbers.