Word problems on equations

Equa­tions become really use­ful when they answer ques­tions writ­ten in words. "The sum of two inte­gers is $-34$. One of them is 55. What is the other?" You can turn a sen­tence like this into an equa­tion, solve it, and check the answer. This les­son shows you how, using sums, dif­fer­ences and prod­ucts of inte­gers.

Three steps for every word prob­lem#

1. Name the unknown. Write "Let the other num­ber be $x$".
2. Write the equa­tion. Turn the sen­tence into sym­bols, using the key words: sum means add, prod­uct means mul­ti­ply, and so on.
3. Solve and check. Solve the equa­tion, then check that the answer fits the words.

Sums of inte­gers#

Exam­ple 1#

The sum of two inte­gers is $-34$. If one of them is 55, find the other.

Let the other inte­ger be $x$. Their sum is $-34$:

$$\begin{aligned}x + 55 &= -34 \\ \Rightarrow x &= -34 - 55 \\ \Rightarrow x &= -89\end{aligned}$$

For $-34 - 55$: both signs are minus, so add $34 + 55 = 89$ and keep the minus.

Check. $-89 + 55$: the signs dif­fer, the dif­fer­ence is $89 - 55 = 34$, and 89 is neg­a­tive. So the sum is $-34$. The other inte­ger is $-89$.

Exam­ple 2#

The sum of two inte­gers is 1982. If one of them is 1947, find the other.

$$\begin{aligned}x + 1947 &= 1982 \\ \Rightarrow x &= 1982 - 1947 \\ \Rightarrow x &= 35\end{aligned}$$

Check: $35 + 1947 = 1982$.

Adding to a num­ber#

Exam­ple 3#

What num­ber should be added to 57 to get $-75$?

Let the num­ber be $x$. Adding it to 57 gives $-75$:

$$\begin{aligned}57 + x &= -75 \\ \Rightarrow x &= -75 - 57 \\ \Rightarrow x &= -132\end{aligned}$$

Check: $57 + (-132) = 57 - 132 = -75$.

Sub­tract­ing from a num­ber#

Exam­ple 4#

What num­ber should be sub­tracted from $-6$ to get $-5$?

"Sub­tracted from $-6$" means we start at $-6$ and take $x$ away:

$$\begin{aligned}-6 - x &= -5 \\ \Rightarrow -x &= -5 + 6 \\ \Rightarrow -x &= 1 \\ \Rightarrow x &= -1\end{aligned}$$

If $-x = 1$, then $x$ is the oppo­site of 1, which is $-1$.

Check. $-6 - (-1) = -6 + 1 = -5$. Tak­ing away a neg­a­tive is the same as adding.

Prod­ucts of inte­gers#

Exam­ple 5#

The prod­uct of two inte­gers is 154. If one of them is $-14$, find the other.

Let the other inte­ger be $x$. Their prod­uct is 154:

$$\displaystyle \begin{aligned}x \times (-14) &= 154 \\ \Rightarrow x &= \frac{154}{-14} \\ \Rightarrow x &= -11\end{aligned}$$

First the num­bers: $154 \div 14 = 11$. Then the sign: a pos­i­tive divided by a neg­a­tive is neg­a­tive.

Check. $(-11) \times (-14) = 154$. Two neg­a­tives give a pos­i­tive.

A prob­lem from every­day life#

Exam­ple 6#

At 6 a.m. the tem­per­a­ture in Leh was $-7$ degrees. By noon it was 5 degrees. By how many degrees did it rise?

Let the rise be $x$ degrees.

$$\begin{aligned}-7 + x &= 5 \\ \Rightarrow x &= 5 + 7 \\ \Rightarrow x &= 12\end{aligned}$$

The tem­per­a­ture rose by 12 degrees.

Exam­ple 7#

Sita had some money. She spent ₹350 and has ₹1200 left. How much did she have at first?

Let the amount she had at first be $x$ rupees.

$$\begin{aligned}x - 350 &= 1200 \\ \Rightarrow x &= 1200 + 350 \\ \Rightarrow x &= 1550\end{aligned}$$

She had ₹1550 at first.

Remem­ber. Write a sen­tence say­ing what $x$ stands for, and fin­ish with a sen­tence that answers the ques­tion.

Com­mon mis­takes#

• Mix­ing up "sub­tract from". "Sub­tract $x$ from $-6$" is $-6 - x$, not $x - (-6)$.
• Sign slips. $-34 - 55$ is $-89$, not $-21$ or $21$.
• For­get­ting the sign rule in prod­ucts. If the prod­uct is pos­i­tive and one num­ber is neg­a­tive, the other is neg­a­tive.
• Not answer­ing the ques­tion. Fin­ish with a sen­tence such as "The other inte­ger is $-89$".

Key terms#

• Sum: the answer when num­bers are added.
• Prod­uct: the answer when num­bers are mul­ti­plied.
• Unknown: the num­ber you are try­ing to find, writ­ten as $x$.

Prac­tice#

1. The sum of two inte­gers is 121. If one of them is 29, find the other.
2. The sum of two inte­gers is $-50$. If one of them is $-23$, find the other.
3. What num­ber should be added to $-19$ to get 6?
4. What num­ber should be sub­tracted from 10 to get 25?
5. When $-40$ is sub­tracted from a num­ber, the result is 18. Find the num­ber.
6. The prod­uct of two inte­gers is 105. If one of them is $-7$, find the other.
7. The prod­uct of two inte­gers is $-96$. If one of them is 8, find the other.
8. A num­ber divided by $-4$ gives 9. Find the num­ber.
9. Twice a num­ber, decreased by 9, gives $-25$. Find the num­ber.
10. The sum of a num­ber and three times the num­ber is $-48$. Find the num­ber.
11. A diver is at $-18$ metres. She rises some metres and reaches $-5$ metres. How far did she rise?
12. Ravi had some money. He spent ₹275 and has ₹980 left. How much did he have at first?

Answers#