Adding and taking away decimals

Dec­i­mals are every­where once you start look­ing. A shop bill says ₹$45.50$. A cricket com­men­ta­tor says a bowler's econ­omy is $6.25$. Your height on the school card is $1.42$ metres. Sooner or later you have to add two of these, or take one away from another. This les­son shows you how to do both with­out a sin­gle slip.

A dec­i­mal is a num­ber with a dot in it. A price like $23.80$ rupees is a dec­i­mal. The dot splits the whole rupees on its left from the part of a rupee on its right. That dot is called the dec­i­mal point.

You add dec­i­mals almost the way you add whole num­bers. There is one extra care to take. The dec­i­mal points must sit in a straight line, one under the other.

First, the same form

The first rule says to change the dec­i­mals into the same form. That means giv­ing them the same num­ber of dig­its after the point.

A digit is a sin­gle num­ber sign. In $23.80$ the 2, the 3, the 8 and the 0 are all dig­its.

Look at $23.8$ and $212.62$. One has a sin­gle digit after the point. The other has two. Write $23.8$ as $23.80$ and now both have two.

Adding that zero is safe. A zero added after the dec­i­mal point, at the right hand end, adds noth­ing to the value.

Think of rupees. There are 100 paise in a rupee. So one paisa is one hun­dredth of a rupee.

Now read $23.8$ rupees. It means 23 rupees and 8 tenths of a rupee. Eight tenths of a rupee is 80 paise. So $23.80$ names the very same money.

Remem­ber. A zero writ­ten after the dec­i­mal point, at the right hand end, never changes the value. So $23.8$ and $23.80$ are the same num­ber.

Line up the points, not the last dig­its

This is the one place where sums go wrong. Do not line up the last dig­its. Line up the dec­i­mal points.

Where a digit sits tells you how much it is worth. That is its place value. In $23.80$ the 8 sits in the tenths place. So it is worth 8 tenths of a rupee, which is 80 paise.

Here is why the points mat­ter. Look at the table below. Each up and down col­umn holds one place value. Tenths go under tenths. Hun­dredths go under hun­dredths.

The two points stand in the same col­umn. That is what you want on paper.

Say you line up the last dig­its instead. Then the 8 tenths of $23.8$ lands under the 2 hun­dredths. You would be adding 80 paise to 2 paise as if they matched. The answer would be wrong.

Adding dec­i­mals

Three things to do every time.

1. Change the given dec­i­mals into the same form.
2. Write dig­its of the same place value under each other, in a col­umn.
3. Add as you add whole num­bers. Then put the dec­i­mal point in the right place.

The point in the answer goes straight below the other points. It does not move. Here is the first sum set out in columns.

The pic­ture below shows the same sum as a place value chart. Look at the pink strip in the mid­dle. Every dec­i­mal point sits in it, includ­ing the point in the answer.

Start at the right, just as you do with whole num­bers. Hun­dredths first: $2 + 0 = 2$. Then tenths: $6 + 8 = 14$ tenths. Four­teen tenths is one whole and four tenths, so write $4$ and carry $1$ into the ones col­umn. Ones: $2 + 3 + 1 = 6$. Tens: $1 + 2 = 3$. Hun­dreds: $2$. Put the point under the other points and read the answer, $236.42$.

Add 212.62 and 23.8

$$\begin{aligned}&212.62 + 23.8 \\ &= 212.62 + 23.80 \\ &= 236.42\end{aligned}$$

The next sum has three num­bers. The longest one has three dig­its after the point. So give all three num­bers three dig­its after the point.

Add 3.625, 4.76 and 35.74

$$\begin{aligned}&3.625 + 4.76 + 35.74 \\ &= 3.625 + 4.760 + 35.740 \\ &= 44.125\end{aligned}$$

Two more sums to try with a pen­cil

Here are two more addi­tions, a lit­tle eas­ier than the ones above. The first needs no car­ry­ing. The sec­ond needs a zero and a carry.

Add 4.5 and 2.3

$$\begin{aligned}&4.5 + 2.3 \\ &= 6.8\end{aligned}$$

Both num­bers already have one digit after the point. Tenths: $5 + 3 = 8$. Ones: $4 + 2 = 6$. Noth­ing to carry.

Add 7.25 and 0.8

$$\begin{aligned}&7.25 + 0.8 \\ &= 7.25 + 0.80 \\ &= 8.05\end{aligned}$$

Hun­dredths: $5 + 0 = 5$. Tenths: $2 + 8 = 10$ tenths, which is one whole and no tenths. Write $0$ and carry $1$. Ones: $7 + 0 + 1 = 8$. The $0$ in the tenths place must stay. With­out it, the answer would wrongly read $8.5$.

Tak­ing away dec­i­mals

Tak­ing away is also called sub­tract­ing. There are four things to do every time.

1. Change the given dec­i­mals into the same form.
2. Write the smaller num­ber below the big­ger num­ber.
3. Write dig­its of the same place value under each other, in a col­umn.
4. Take away as you do with whole num­bers.

The big­ger num­ber goes on top. You can­not take 8 rupees away from 3 rupees. Put the big­ger num­ber first and every col­umn has enough to give. The dec­i­mal points still line up.

Sub­tract 42.32 from 83.95

$$\begin{aligned}&83.95 - 42.32 \\ &= 41.63\end{aligned}$$

Both num­bers already have two dig­its after the point. So there is no zero to add here.

When one num­ber has no point

A whole num­ber has no point writ­ten down. You may put one at its right hand end. So 40 is $40.0$, and 100 is $100.00$.

Think of money again. 40 rupees is 40 rupees and no paise. So $40.0$ is the same amount as 40.

Now the top num­ber has the same form as the one below it. You can take away col­umn by col­umn.

Take 13.6 away from 40

$$\begin{aligned}&40 - 13.6 \\ &= 40.0 - 13.6 \\ &= 26.4\end{aligned}$$

You can also see this sum on a num­ber line. Start at $40$ and take away $13.6$ in three easy jumps: first $10$, then $3$, then $0.6$.

Tak­ing away with bor­row­ing

Some­times the top digit in a col­umn is smaller than the one below it. Then you bor­row from the col­umn on the left, just as you do with whole num­bers. The dec­i­mal point does not get in the way. You bor­row straight across it.

Sub­tract 2.75 from 5.6

$$\begin{aligned}&5.6 - 2.75 \\ &= 5.60 - 2.75 \\ &= 2.85\end{aligned}$$

Hun­dredths: $0 - 5$ can­not be done, so bor­row one tenth. Now it is $10 - 5 = 5$. Tenths: the $6$ has become $5$, and $5 - 7$ can­not be done, so bor­row one whole from the ones. Now it is $15 - 7 = 8$. Ones: the $5$ has become $4$, and $4 - 2 = 2$. The answer is $2.85$.

Sub­tract 18.45 from 50

$$\begin{aligned}&50 - 18.45 \\ &= 50.00 - 18.45 \\ &= 31.55\end{aligned}$$

Here the top num­ber is writ­ten with two zeros after the point, because $18.45$ has two dig­its after its point. Every col­umn now has a digit to work with.

A sum with adding and tak­ing away

Some sums mix the two. Gather the num­bers you are adding. Gather the num­bers you are tak­ing away. Then do one tak­ing away at the end.

Here is why that is allowed. Think of money com­ing in and money going out. Add up every­thing com­ing in. Add up every­thing going out. Then take the going out total away from the com­ing in total. The order you count them in does not change what is left.

In the sum below, $123.52 + 2353.254 = 2476.774$. The num­bers being taken away give $27.53 + 167.2345 = 194.7645$.

The last step needs the same form. So $2476.774$ is writ­ten as $2476.7740$ first.

Find the value of 123.52 - 27.53 + 2353.254 - 167.2345

$$\begin{aligned}&123.52 - 27.53 + 2353.254 - 167.2345 \\ &= 2476.774 - 194.7645 \\ &= 2476.7740 - 194.7645 \\ &= 2282.0095\end{aligned}$$

Remem­ber. Line up the dec­i­mal points. Fill any short num­ber with zeros after its dec­i­mal point. Then work as you do with whole num­bers.

A shop­ping sum

Meena buys a note­book for ₹$12.50$ and a pen for ₹$3.75$. She pays with a ₹$100$ note. How much change does she get?

$$\begin{aligned}\text{Total} &= 12.50 + 3.75 = 16.25 \\ \text{Change} &= 100.00 - 16.25 = 83.75\end{aligned}$$

Meena gets ₹$83.75$ back. Check it by adding: $16.25 + 83.75 = 100.00$. Adding back is the quick­est way to check any tak­ing away.

Prac­tice

Com­mon mis­takes

• Lin­ing up the last dig­its instead of the dec­i­mal points. In $23.8 + 212.62$ this puts tenths under hun­dredths.
• Drop­ping a zero that holds a place. $8.05$ and $8.5$ are dif­fer­ent num­bers.
• For­get­ting that a whole num­ber has a point at its right-hand end. $40$ is $40.0$, not $0.40$.
• Putting the smaller num­ber on top when tak­ing away, and then tak­ing the top digit from the bot­tom one.
• Leav­ing the dec­i­mal point out of the answer, or mov­ing it. It goes straight below the other points.

Key terms

Dec­i­mal

A num­ber with a dec­i­mal point, such as $23.80$, that shows parts of a whole.

Dec­i­mal point

The dot that sep­a­rates the whole part on its left from the part less than one on its right.

Digit

A sin­gle num­ber sign from $0$ to $9$.

Place value

What a digit is worth because of where it sits: ones, tenths, hun­dredths and so on.

Same form

Dec­i­mals writ­ten with the same num­ber of dig­its after the point, by adding zeros at the right-hand end.

Carry

Mov­ing a ten from one col­umn into the next col­umn to the left when a col­umn adds to ten or more.

Bor­row

Tak­ing one from the col­umn to the left when a top digit is too small to take away from.

Answers

1. $6.4 + 3.75 = 6.40 + 3.75 = 10.15$
2. $15.2 + 0.68 = 15.20 + 0.68 = 15.88$
3. $2.5 + 0.75 + 1.8 = 2.50 + 0.75 + 1.80 = 5.05$
4. $9.5 - 2.35 = 9.50 - 2.35 = 7.15$
5. $40 - 13.6 = 40.0 - 13.6 = 26.4$
6. $100 - 45.25 = 100.00 - 45.25 = 54.75$