Word problems on equations

Equa­tions are most use­ful when a prob­lem is given in words. Prices, lengths, ages and shares can all be turned into an equa­tion, and the equa­tion does the hard think­ing for you. In this les­son you will learn a few steps that turn words into an equa­tion, and then use them on prob­lems with frac­tions, dec­i­mals and money.

Five steps#

1. Read the prob­lem care­fully. Find what you are asked to find.
2. Let the unknown num­ber be $x$. Write down what $x$ stands for.
3. Write the other amounts using $x$.
4. Form the equa­tion from the words, and solve it.
5. Check the answer in the words of the prob­lem, then write the answer as a sen­tence.

Some words to watch: "sum" means add, "dif­fer­ence" means take away, "prod­uct" means mul­ti­ply, and "is" or "gives" usu­ally becomes the equals sign.

Find­ing a miss­ing num­ber#

Exam­ple 1#

The sum of two ratio­nal num­bers is $\displaystyle -\frac{5}{6}$. One of them is $\displaystyle \frac{1}{3}$. Find the other.

Let the other num­ber be $x$.

$$\displaystyle \begin{aligned} x + \frac{1}{3} &= -\frac{5}{6} \\ \Rightarrow x &= -\frac{5}{6} - \frac{2}{6} \\ \Rightarrow x &= -\frac{7}{6} \end{aligned}$$

The other num­ber is $\displaystyle -\frac{7}{6}$.

Exam­ple 2#

What num­ber should be added to $\displaystyle \frac{3}{4}$ to get $\displaystyle -\frac{1}{2}$?

Let the num­ber be $x$. Then $\displaystyle \frac{3}{4} + x = -\frac{1}{2}$.

$$\displaystyle x = -\frac{1}{2} - \frac{3}{4} = -\frac{2}{4} - \frac{3}{4} = -\frac{5}{4}$$

Exam­ple 3#

What num­ber should be sub­tracted from $2.5$ to get $4.1$?

Let the num­ber be $x$. "Sub­tracted from $2.5$" means $2.5 - x$.

$$\begin{aligned} 2.5 - x &= 4.1 \\ \Rightarrow -x &= 4.1 - 2.5 = 1.6 \\ \Rightarrow x &= -1.6 \end{aligned}$$

Check: $2.5 - (-1.6) = 2.5 + 1.6 = 4.1$. It works.

Exam­ple 4#

The prod­uct of two ratio­nal num­bers is $\displaystyle -\frac{9}{10}$. One of them is $\displaystyle \frac{3}{5}$. Find the other.

Let the other be $x$. Then $\displaystyle x \times \frac{3}{5} = -\frac{9}{10}$.

$$\displaystyle x = -\frac{9}{10} \times \frac{5}{3} = -\frac{45}{30} = -\frac{3}{2}$$

Parts of a num­ber#

Exam­ple 5#

A num­ber added to half of itself gives $21$. Find the num­ber.

Let the num­ber be $x$. Half of it is $\displaystyle \frac{x}{2}$.

$$\displaystyle \begin{aligned} x + \frac{x}{2} &= 21 \\ \Rightarrow \frac{3x}{2} &= 21 \\ \Rightarrow 3x &= 42 \\ \Rightarrow x &= 14 \end{aligned}$$

Check: $14 + 7 = 21$. The num­ber is $14$.

Exam­ple 6#

When one third of a num­ber is added to $5$, the result is $12$. Find the num­ber.

$$\displaystyle \begin{aligned} \frac{x}{3} + 5 &= 12 \\ \Rightarrow \frac{x}{3} &= 7 \\ \Rightarrow x &= 21 \end{aligned}$$

The num­ber is $21$.

Money, lengths and ages#

Exam­ple 7#

Sita has ₹45.50 more than Ravi. Together they have ₹225.50. How much does each have?

Let Ravi have ₹$x$. Then Sita has ₹$(x + 45.50)$.

$$\begin{aligned} x + (x + 45.50) &= 225.50 \\ \Rightarrow 2x &= 225.50 - 45.50 = 180 \\ \Rightarrow x &= 90 \end{aligned}$$

Ravi has ₹90 and Sita has ₹135.50. Check: $90 + 135.50 = 225.50$.

Exam­ple 8#

The length of a rec­tan­gle is $2.5$ m more than its breadth. Its perime­ter is $17$ m. Find the length and the breadth.

Let the breadth be $x$ m. Then the length is $(x + 2.5)$ m. The perime­ter of a rec­tan­gle is $2 \times (\text{length} + \text{breadth})$.

$$\begin{aligned} 2(x + 2.5 + x) &= 17 \\ \Rightarrow 4x + 5 &= 17 \\ \Rightarrow 4x &= 12 \\ \Rightarrow x &= 3 \end{aligned}$$

The breadth is $3$ m and the length is $5.5$ m. Check: $2 \times (5.5 + 3) = 17$.

Exam­ple 9#

A father is $3$ times as old as his son. In $12$ years, he will be twice as old as his son. Find their ages now.

Let the son be $x$ years old. Then the father is $3x$ years old. In $12$ years they will be $x + 12$ and $3x + 12$.

$$\begin{aligned} 3x + 12 &= 2(x + 12) \\ \Rightarrow 3x + 12 &= 2x + 24 \\ \Rightarrow 3x - 2x &= 24 - 12 \\ \Rightarrow x &= 12 \end{aligned}$$

The son is $12$ and the father is $36$. Check: in $12$ years they will be $24$ and $48$, and $48$ is twice $24$.

Remem­ber. Say what $x$ stands for, form the equa­tion from the words, solve it, and check the answer in the story.

Com­mon mis­takes#

• Not say­ing what $x$ stands for. Then the answer has no mean­ing.
• Read­ing "sub­tract $a$ from $b$" as $a - b$. It is $b - a$.
• Giv­ing only $x$ when the ques­tion asks for two amounts. In Exam­ple 7 both Rav­i's and Sita's money are needed.
• For­get­ting units, such as rupees, metres or years, in the final answer.

Prac­tice#

1. The sum of two ratio­nal num­bers is $\displaystyle \frac{7}{8}$. One of them is $\displaystyle -\frac{1}{4}$. Find the other.
2. What num­ber should be added to $\displaystyle -\frac{2}{3}$ to get $\displaystyle \frac{1}{6}$?
3. What num­ber should be sub­tracted from $-1.2$ to get $0.8$?
4. The prod­uct of two ratio­nal num­bers is $\displaystyle \frac{14}{15}$. One of them is $\displaystyle -\frac{7}{5}$. Find the other.
5. A num­ber minus one fourth of itself is $27$. Find the num­ber.
6. Half of a num­ber added to one third of it gives $15$. Find the num­ber.
7. Two fifths of a num­ber is $3.2$ more than one fifth of it. Find the num­ber.
8. A pen costs ₹12.50 more than a pen­cil. Together they cost ₹20. Find the cost of each.
9. The length of a rec­tan­gle is twice its breadth. Its perime­ter is $22.5$ cm. Find the length and the breadth.
10. A mother is $4$ times as old as her daugh­ter. In $6$ years, she will be $3$ times as old as her daugh­ter. Find their ages now.

Answers#