Word Problems with Linear Equations: Practice

Let $n$ be the number. Adding $5$ to it gives $12$, so set up and solve the equation.

$n + 5 = 12, \qquad n = 7$

Check: $7 + 5 = 12$, so the number is $7$.

Let $n$ be the number. Twice the number is $2n$, and it equals $18$.

$2n = 18, \qquad n = 9$

Check: $2 \times 9 = 18$, so the number is $9$.

Let $n$ be the smaller integer, so the next one is $n + 1$. Their sum is $23$.

$n + (n + 1) = 23, \qquad 2n + 1 = 23, \qquad n = 11$

The integers are $11$ and $12$, and $11 + 12 = 23$, so the smaller is $11$.

Let $n$ be the unknown number. The phrase $50\%$ of $n$ means $\frac{1}{2}n$, and it equals $18$.

$\tfrac{1}{2}n = 18, \qquad n = 36$

Check: $50\%$ of $36$ is $18$, so the number is $36$.

The total cost is the number of pencils times the price of each, and each pencil costs $2$ dollars.

$2 \times p = 2p$

So $p$ pencils cost $2p$ dollars.

Consecutive integers step by $1$, so if the smaller is $n$, the next is $n + 1$. Their sum is $47$.

$n + (n + 1) = 47$

The gap between consecutive integers is $1$, which is why the second integer is $n + 1$.

Let $w$ be the width in centimeters, so the length is $2w$. The perimeter is twice the sum of length and width.

$2(2w + w) = 24, \qquad 6w = 24, \qquad w = 4$

Check: with width $4$ and length $8$, the perimeter is $2(8 + 4) = 24$ cm, so the width is $4$ cm.

Let $n$ be the number of students. Then $20\%$ of $n$, which is $\frac{1}{5}n$, equals $6$.

$\tfrac{1}{5}n = 6, \qquad n = 30$

Check: $20\%$ of $30$ is $6$, so there are $30$ students.

Let $x$ be the number of years from now. In $x$ years Ana is $x$ years older, so her age is $12 + x$, and that equals $20$.

$12 + x = 20$

Solving gives $x = 8$, so the equation is $12 + x = 20$.

Let $n$ be the number. The sum of the number and $12$ is $n + 12$, which equals $30$.

$n + 12 = 30, \qquad n = 18$

Check: $18 + 12 = 30$, so the number is $18$.

Let $n$ be the smaller even integer, so the next even integer is $n + 2$. Their sum is $30$.

$n + (n + 2) = 30, \qquad 2n + 2 = 30, \qquad n = 14$

The integers are $14$ and $16$, so the larger is $16$; check $14 + 16 = 30$.

Let $n$ be the number. Four times the number is $4n$, and it equals $32$.

$4n = 32, \qquad n = 8$

Check: $4 \times 8 = 32$, so the number is $8$.