Writing words as symbols

Alge­bra begins with a sim­ple step: turn­ing a sen­tence into sym­bols. A shop­keeper who says "six more than yes­ter­day" or a par­ent who says "half of the sweets" is already talk­ing alge­bra. Once you can write such sen­tences in sym­bols, you can test them with num­bers and, later, solve equa­tions. Lis­ten for the key word, and take just as much care over the order.

Key words#

Each sen­tence hides a word that tells you what to do.

The bar model shows the idea. The long bar is $x$, the unknown num­ber of sweets in a box. Adding makes the bar longer, tak­ing away cuts a piece off, and halv­ing splits it into two equal parts.

Adding#

The sum of $a$ and $b$ is $a + b$.

6 more than $x$ starts at $x$ and goes up by 6. You write $x + 6$.

With adding, the order does not mat­ter: $x + 6$ and $6 + x$ give the same answer. Test with $x = 7$: $7 + 6 = 13$.

Tak­ing away#

Now the order mat­ters a lot. $5 - 3$ is 2, but $3 - 5$ is not the same.

The sub­trac­tion of $b$ from $a$ is $a - b$. The word "from" tells you where to start: you start at $a$ and take $b$ off. For exam­ple, the sub­trac­tion of 3 from 5 is $5 - 3 = 2$.

$x$ decreased by 5 means $x$ is made smaller by 5. You write $x - 5$.

The sub­trac­tion of $2x$ from 3 starts at 3. You write $3 - 2x$.

Remem­ber. Ask which num­ber you start with. Write that num­ber first, and the other one after the minus sign.

Mul­ti­ply­ing#

The prod­uct of 2 and $x$ is $2 \times x$. We leave out the times sign and write $2x$, with the num­ber in front. The sign $\times$ looks too much like the let­ter $x$, so between a num­ber and a let­ter it is dropped.

5 times $x$ is $5x$. Test with $x = 3$: $5 \times 3 = 15$.

5 times $x$ added to 3 times $y$. Five times $x$ is $5x$, and three times $y$ is $3y$. "Added to" means $5x$ is added to $3y$, so you write $3y + 5x$. Since order does not mat­ter in adding, this is the same as $5x + 3y$.

Divid­ing#

Divide $x$ by 2 gives $\displaystyle \frac{x}{2}$. The num­ber being shared out goes on top. The line means the same as $\div$.

40 divided by $x$ gives $\displaystyle \frac{40}{x}$. Here 40 is shared out, so 40 goes on top. Test with $x = 8$: $40 \div 8 = 5$.

Half of $x$ is $\displaystyle \frac{x}{2}$. One third of $x$ is $\displaystyle \frac{x}{3}$.

Exam­ple 1#

Write "4 less than half of $x$" in sym­bols.

Halve first: half of $x$ is $\displaystyle \frac{x}{2}$. Then go 4 below it: $\displaystyle \frac{x}{2} - 4$.

Test with $x = 10$: half of 10 is 5, and 4 less than 5 is 1.

Use­ful facts#

These five facts are used all the time.

Exam­ple 2#

Write $-2 \times a$ and $-1 \times p$ the short way.

$-2 \times a = -2a$, and $-1 \times p = -p$.

The sen­tences together#

Longer sen­tences#

Exam­ple 3#

Write "twice the sum of $x$ and 3" in sym­bols.

The sum of $x$ and 3 is $x + 3$. Twice that is $2(x + 3)$. The brack­ets mat­ter: with­out them, $2x + 3$ would dou­ble only $x$. Test with $x = 4$: $2(4 + 3) = 14$, but $2 \times 4 + 3 = 11$.

Com­mon mis­takes#

• Writ­ing $4 - x$ for "4 less than $x$". You start at $x$, so it is $x - 4$.
• Writ­ing $x5$ instead of $5x$. The num­ber goes in front.
• Turn­ing a divi­sion upside down. "40 divided by $x$" is $\displaystyle \frac{40}{x}$, not $\displaystyle \frac{x}{40}$.
• Writ­ing $1x$. One lot of $x$ is writ­ten $x$.

Key terms#

• Sym­bol: a short mark that stands for some­thing longer, such as $+$ or a let­ter.
• Sum: the answer when you add.
• Prod­uct: the answer when you mul­ti­ply.
• Expres­sion: num­bers and let­ters joined by oper­a­tion signs, such as $3y + 5x$.

Prac­tice#

1. Write in sym­bols: the sum of $m$ and $n$.
2. Write in sym­bols: 9 more than $y$.
3. Write in sym­bols: the sub­trac­tion of $y$ from $p$.
4. Write in sym­bols: $k$ decreased by 7.
5. Write in sym­bols: the prod­uct of 7 and $k$.
6. Write in sym­bols: divide $p$ by 4.
7. Write in sym­bols: 12 divided by $t$.
8. Write in sym­bols: one third of $n$.
9. Write in sym­bols: 2 times $a$ added to 5 times $b$.
10. Write in sym­bols: 6 less than half of $n$.
11. Write $-2 \times a$ and $-1 \times p$ the short way.
12. If $x = 12$, find the value of $\displaystyle \frac{x}{2} - 4$.

Answers#