Writing words as symbols

Alge­bra begins with a sim­ple step: turn­ing a sen­tence into sym­bols. A shop­keeper who says "three more than yes­ter­day" or a teacher who says "half the class" is already talk­ing alge­bra. Once you can write such sen­tences in sym­bols, you can work with them, test them with num­bers, and later solve equa­tions. This les­son teaches the key words to lis­ten for and, just as impor­tant, the order to write things in.

Get­ting started

Imag­ine a box of sweets with the lid shut. You want to talk about how many are inside. But you can­not count them.

A box of sweets

So give that num­ber a name. Call it $x$. The let­ter $x$ is not a sweet. It is the num­ber of sweets. If the box holds $7$ sweets, then $x$ is $7$. If it holds $12$, then $x$ is $12$. The let­ter sim­ply waits.

Now pic­ture a bag you have not weighed. Call its weight $w$. You can still say true things about it. Two such bags weigh $2w$. Now put $3$ kg into the bag. The bag is heav­ier than before. It weighs $w + 3$. You did all that with no scales.

Some sym­bols never change. The num­ber $5$ is always $5$. A sym­bol with one fixed value is called a con­stant.

Remem­ber. The same let­ter can stand for a dif­fer­ent num­ber in a dif­fer­ent ques­tion. Today $x$ is $7$. Tomor­row $x$ might be $12$. A sym­bol that can take var­i­ous num­ber val­ues is called a vari­able. Another name for it is a lit­eral.

From words to sym­bols

A sym­bol is a short mark that stands for some­thing longer. The mark $+$ is a sym­bol. It stands for the word add. A let­ter is a sym­bol too. It stands for a num­ber.

A ques­tion often comes to you as a sen­tence in words. Your job is to write it in sym­bols. The sym­bols say the same thing in less room.

Each of these sen­tences hides a word that tells you what to do. A few hide two. Find them all first. They tell you whether to add, take away, mul­ti­ply or divide. Then sort out the order.

A bar model is a good way to pic­ture these phrases. The long bar stands for $x$, the unknown num­ber of sweets. Adding makes the bar longer; tak­ing away cuts a piece off; halv­ing splits it into two equal parts.

Words that mean add

The word sum means add. So the sum of $a$ and $b$ is $a + b$.

More than also means add. Six more than $x$ starts at $x$ and goes up by $6$. You write $x + 6$.

Added to means add as well. With adding, the order does not mat­ter.

Here is why. Put $7$ sweets on the table. Then put $6$ more beside them. Count them all. Now clear them away and start again. This time put $6$ sweets down first. Then put $7$ more beside them. Count again. You get $13$ both times. Mov­ing them round does not change how many there are. That is why $x + 6$ and $6 + x$ give the same answer.

A let­ter can take var­i­ous num­ber val­ues. So you can always test your sym­bols by putting a num­ber in. Say $x$ is $7$. Then $x + 6$ is $13$. Seven sweets and six more make thir­teen. That matches.

Words that mean take away

Three groups of words mean take away. They are sub­trac­tion of, decreased by, and less than.

Order mat­ters now, and it mat­ters a lot. Say the box holds $5$ sweets and you eat $2$. Five take away two leaves $3$. Now try it the other way round. You have $2$ sweets. Some­one asks you to take $5$ away. You can­not do it. There are not enough. So $5 - 2$ and $2 - 5$ are not the same thing. That is why you must get the first num­ber right.

Remem­ber. Say the sen­tence slowly. Ask which num­ber you start with. Write that num­ber first. The other one goes after the minus sign.

Take the sub­trac­tion of $b$ from $a$. The word from tells you where you begin. You begin at $a$. You take $b$ off it. So you write $a - b$.

Find­ing the order

$$\begin{aligned}&\text{Sentence: the subtraction of } b \text{ from } a \\ &\text{You start at } a \\ &\text{You take away } b \\ &\text{In symbols: } a - b\end{aligned}$$

Now take $5$ decreased by $x$. Decreased means made smaller. The thing being made smaller is $5$. So you start at $5$ and take $x$ off. You write $5 - x$.

This one is easy to get the wrong way round. It is tempt­ing to write $x - 5$, because $x$ is the last thing you read. But that would mean you start at $x$. The sen­tence starts at $5$.

Four less than $x$ sounds as if $4$ comes first. Ask your­self: less than what? The sen­tence says less than $x$. So $x$ is the num­ber you start at. You go $4$ below it. Pic­ture a stair with $x$ writ­ten on it. Four less than $x$ means step down four stairs. You write $x - 4$.

Now com­pare that with $5$ decreased by $x$. There the sen­tence names the thing being made smaller. That is the $5$. So you have two small tests. The word after than is where you start. The word before decreased is where you start.

Here is one more like it. The sub­trac­tion of $2x$ from $3$ begins at $3$. You take $2x$ off it. So you write $3 - 2x$.

Words that mean mul­ti­ply

Prod­uct and times both mean mul­ti­ply.

The prod­uct of $2$ and $x$ means $2 \times x$. You write it as $2x$. The mul­ti­pli­ca­tion sign is left out.

There is a good rea­son for drop­ping that sign. The sign $\times$ looks very like the let­ter $x$. Writ­ten side by side they are hard to tell apart. So maths leaves the sign out between a num­ber and a let­ter.

The num­ber goes in front of the let­ter. You write $5x$, not $x5$. Every­one writes it the same way, so every­one can read it.

So $5$ times $x$ is $5x$. Test it with a num­ber. Say $x$ is $3$. Then $5x$ is $5 \times 3$, which is $15$.

When there are two let­ters

Some­times there are two num­bers you do not know. Then you need two let­ters. Call the first one $x$ and the sec­ond one $y$. They are dif­fer­ent let­ters because they are dif­fer­ent num­bers.

Now take $5$ times $x$ added to $3$ times $y$. Five times $x$ is $5x$. Three times $y$ is $3y$. Added to means add. So you write $5x + 3y$.

Words that mean divide

Divid­ing means shar­ing into equal piles. Take the box of sweets and share them between $2$ chil­dren. Each child gets half of what was in the box. You write that as $\displaystyle \frac{x}{2}$.

The num­ber being shared out sits on top. The num­ber of piles sits under­neath. The line does the same job as the divide sign. So $\displaystyle \frac{x}{2}$ means the same as $x \div 2$.

Divided by and half of both mean divide. So divide $x$ by $2$ gives $\displaystyle \frac{x}{2}$.

Forty divided by $x$ gives $\displaystyle \frac{40}{x}$. Here $40$ is the num­ber being shared out. So $40$ goes on top.

Order mat­ters in divi­sion too. $\displaystyle \frac{x}{2}$ and $\displaystyle \frac{2}{x}$ are not the same. Ask which num­ber is being shared out. That one goes on top.

Test that with a num­ber. Say $x$ is $8$. Then $\displaystyle \frac{40}{x}$ is $40 \div 8$, which is $5$.

Half of a num­ber means divide it by $2$. So half of $x$ is $\displaystyle \frac{x}{2}$.

This next one needs two steps. Four less than half of $x$ means you halve first. Then you take $4$ away from what you get.

Halv­ing first, then tak­ing away

$$\displaystyle \begin{aligned}&\text{Sentence: } 4 \text{ less than half of } x \\ &\text{Half of } x \text{ is } \frac{x}{2} \\ &\text{You start at } \frac{x}{2} \\ &\text{You take away } 4 \\ &\text{In symbols: } \frac{x}{2} - 4\end{aligned}$$

Try it with a num­ber. Say $x$ is $10$. Half of $10$ is $5$. Then take $4$ from $5$, which leaves $1$.

Longer sen­tences

Real ques­tions often join two oper­a­tions. Work them out in the order the sen­tence describes, and use brack­ets when a whole amount is being mul­ti­plied.

Five more than three times $x$

$$\begin{aligned}&\text{Three times } x \text{ is } 3x \\ &\text{Five more than that: } 3x + 5\end{aligned}$$

Test with $x = 4$: $3 \times 4 + 5 = 17$.

Twice the sum of $x$ and $3$

$$\begin{aligned}&\text{The sum of } x \text{ and } 3 \text{ is } x + 3 \\ &\text{Twice that: } 2(x + 3)\end{aligned}$$

The brack­ets mat­ter. With­out them, $2x + 3$ would dou­ble only $x$. Test with $x = 4$: $2(4 + 3) = 14$, but $2 \times 4 + 3 = 11$.

Two more than one-third of $x$

One-third of $x$ is $\displaystyle \frac{x}{3}$, so the answer is $\displaystyle \frac{x}{3} + 2$. With $x = 9$ this is $3 + 2 = 5$.

$3$ times $x$ sub­tracted from $10$

You start at $10$ and take $3x$ away, so you write $10 - 3x$. With $x = 2$ this is $10 - 6 = 4$.

The prod­uct of $x$ and $y$

Two let­ters side by side mean mul­ti­ply, so you write $xy$. With $x = 6$ and $y = 4$ this is $24$.

The eleven sen­tences together

Read­ing sym­bols back into words

Going the other way is just as use­ful. You look at sym­bols and say them in words. It shows you have really under­stood.

Prac­tice

These are your turn to try. For each one, first find the key word, then decide which num­ber you start with, and only then write the sym­bols. When you fin­ish, put a small num­ber in place of the let­ter and check that your sym­bols give the same result as the sen­tence.

Write each sen­tence in sym­bols. Say it slowly first. Ask which num­ber you start with.

1. $\text{The sum of } m \text{ and } n$
2. $9 \text{ more than } y$
3. $\text{The product of } 7 \text{ and } k$
4. $\text{Divide } p \text{ by } 4$
5. $8 \text{ decreased by } y$
6. $\text{The subtraction of } q \text{ from } p$
7. $12 \text{ divided by } t$
8. $3 \text{ less than } h$
9. $2 \text{ times } a \text{ added to } 5 \text{ times } b$
10. $6 \text{ less than half of } n$

Check your work here. Every answer is also writ­ten out in the Answers sec­tion at the end.

Remem­ber. The word tells you what to do. The order tells you which num­ber comes first. Get both right and your sym­bols are right.

Com­mon mis­takes

• Writ­ing $4 - x$ for "4 less than $x$". The num­ber after "than" is where you start, so it is $x - 4$.
• Writ­ing $x - 5$ for "5 decreased by $x$". The thing being decreased is $5$.
• 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 $\displaystyle \frac{x}{40}$.
• Leav­ing out brack­ets, so that "twice the sum of $x$ and $3$" becomes $2x + 3$.

Key terms

Sym­bol

A short mark that stands for some­thing longer, such as $+$ or a let­ter.

Vari­able (lit­eral)

A sym­bol that can take var­i­ous num­ber val­ues.

Con­stant

A sym­bol with one fixed value, such as $5$.

Sum

The answer when you add.

Prod­uct

The answer when you mul­ti­ply.

Expres­sion

Num­bers and let­ters joined by oper­a­tion signs, such as $5x + 3y$.

Answers

1. $m + n$
2. $y + 9$
3. $7k$
4. $\displaystyle \frac{p}{4}$
5. $8 - y$
6. $p - q$
7. $\displaystyle \frac{12}{t}$
8. $h - 3$
9. $2a + 5b$
10. $\displaystyle \frac{n}{2} - 6$