Word Problems

Word problems relate mathematical formulas and methods to real world problems. On this page we give a sampling of some typical word problems. We provide further explanations of word problems here.

Discount/Markup

A furniture store is having a sale where all tables are 15% off. If the non-sale price of a table is $180 and the sales tax rate is 5%, what is the total cost of the table during the sale? a.$156.30
b. $158.35 c.$160.65
d. $163.85 e.$166.55

Solution:

To figure out the total cost of the table, multiply 180 by .85 to get the sale price. Then multiply that price by the sales tax percentage to get the amount that needs to be added to the sale price.

\begin{align}180 \times .85 = 153\end{align}

\begin{align}153 \times .05 = 7.65\end{align}

\begin{align}153 + 7.65 =160.65\end{align}

Geometry

A rectangular garden has a length of 15 feet and a width of 8 feet. A path of width x surrounds the garden and has an area of 108 square feet. How many feet is the outside perimeter of the path?

a. 62
b. 70
c. 92
d. 108
e. 124

Solution:

To solve this problem, we’ll set up an equation to determine the dimensions of the path with x equaling the width of the path.

\begin{align}(15 + 2x)(8 + 2x) = (15 \times 8) + 108\end{align}

Notice that both the left side and the right side of this equation compute the total area of the garden, including the path. On the right side of the equation, the area of the garden inside the path is 15 x 8, while the area of the path itself is 108. As to the left side of the equation: Since the path is of width x, and it completely surrounds the garden, then we must add 2x to both dimensions to get the full area of the garden, including the path.

Now solve for x.

\begin{align}120 + 30x + 16x + 4x^2 = 120 + 108\end{align}

\begin{align}4x^2 + 46x – 108 = 0\end{align}

\begin{align}2x^2 + 23x – 54 = 0\end{align} (Previous equation divided by 2)

\begin{align}(2x + 27)(x – 2) = 0\end{align}

The two possible values for x are -13.5 and 2, but, since the width of the path can’t be negative, we can assume that the width is 2 feet. We’ll now add this to the dimensions of the garden to get the perimeter.

\begin{align}2(15 + (2 \times 2)) + 2(8 + (2 \times 2)) =\end{align}

\begin{align}38 + 24 = 62\end{align}

Interest and Interest Rates

Oscar invested a certain amount of money at 12% simple interest, and he invested an amount of money at 9% simple interest that is ¾ of the amount that he invested at 12%. After 1 year, the total amount of interest from both investments is $138. How much did he invest altogether? a.$1288
b. $1376 c.$1464
d. $1542 e.$1692

Solution:

To solve this problem, we’ll set up an equation with x equaling the amount of money he invested at 12%.

\begin{align}.12x + .09(3x/4) = 138\end{align}

Solve for x.

\begin{align}.12x + .27x/4 = 138\end{align}

\begin{align}.48x/4 + .27x/4 = 138\end{align}

\begin{align}.75x/4 = 138\end{align}

\begin{align}x = 138/(.75/4) = (138 \times 4)/.75 = 552/.75 = 736\end{align}

Now we add to this value the amount of money he invested at 9%.

\begin{align}736 + 3/4(736) = 736 + 2208/4 = 736 + 552 = 1288\end{align}

Mixtures

How many ounces of a 37% salt solution should be mixed with a 13% salt solution to get 68 ounces of a 31% salt solution?

a. 36
b. 40
c. 44
d. 48
e. 51

Solution:

To solve this problem,, we’ll set up an equation with x equaling the amount of the 37% solution that needs to be added. We use the formula

\begin{align}NewPercent = \frac{Percent1 \times Quantity1 + Percent 2 \times Quantity2}{Quantity1 + Quantity2}\end{align}

\begin{align}.31 = \frac{.37x + .13(68 – x)}{68}\end{align}

Solve for x.

\begin{align}.31(68) = .37x + .13(68) – .13x\end{align}

\begin{align}21.08 – 8.84 = .37x – .13x = .24x\end{align}

\begin{align}\frac{12.24}{.24} = x\end{align}

\begin{align}51 = x\end{align}

Distance/Rate/Time

Two trains leave towns 240 miles apart at the same time, traveling towards each other. Train A is traveling 41 miles per hour and train B is traveling 34 miles per hour. How many hours will it take them to meet each other?

a. 3.2
b. 3.8
c. 4.1
d. 4.5
e. 5.4

Solution:

To solve this problem, we’ll use a formula to determine time, which is distance divided by rate of speed. Since the trains are traveling towards each other, the rate by which the distance is being traveled is the sum of their speeds.

\begin{align}\frac {240}{41 + 34} = \frac {240}{75} = \frac {16}{5} = 3.2\end{align}

20 workers start a project that will take them 36 days to complete. After 11 days, 8 of them stop working and the others continue for 5 more days before additional workers are hired. How many new workers need to be hired at that point in order for the project to be done in the original estimated number of days?

a. 9
b. 10
c. 11
d. 12
e. 13

Solution:

To solve this problem, we need to first determine how many worker-days this project requires. We get this by multiplying the number of workers by the number of allocated days.

\begin{align}(20 \times 11) + (20 – 8) \times 5 + (36 – 11 – 5)x = 720\end{align}

Solve for x.

\begin{align}220 + 60 + 20x = 720\end{align}

\begin{align}280 + 20x = 720\end{align}

\begin{align}20x = 720 – 280 = 440\end{align}

\begin{align}x = \frac {440}{20} = 22\end{align}

Since the question asked how many workers needed to be hired, we need to subtract the number already working from this number.

\begin{align}22 – (20 – 8) = 10\end{align}

Unit Conversion

If an object is going at a speed of 1,070 feet per minute, how many inches per second is it traveling?

a. 150
b. 161
c. 182
d. 193
e. 214

Solution:

We’ll convert feet per minute into inches per minute by multiplying the number of feet per minute by 12.

\begin{align}1,070 \times 12 = 12,840\end{align}

We’ll convert inches per minute into inches per second by dividing the number of inches per minute by 60.

\begin{align}12,840 / 60 = 214\end{align}