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'll need to add two of these, or take one away from another, and you'll want to do it with­out a sin­gle slip.

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

Good news: you add dec­i­mals almost exactly the way you add whole num­bers. There is just one extra thing to watch. The dec­i­mal points must sit in a straight line, one under the other.

First, the same form#

The first rule is to change the dec­i­mals into the same form, which sim­ply 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 and the other has two. Write $23.8$ as $23.80$, and now both have two.

Is adding that zero safe? Yes. A zero added after the dec­i­mal point, at the right hand end, adds noth­ing to the value.

Think of rupees to con­vince your­self. 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, and 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#

If dec­i­mal sums go wrong, this is almost always where. Don't line up the last dig­its; line up the dec­i­mal points.

Where a digit sits tells you how much it is worth, and 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.

So why do the points mat­ter so much? Look at the table below. Each up-and-down col­umn holds one place value: tenths go under tenths, and hun­dredths go under hun­dredths.

The two points stand in the same col­umn, and that is exactly what you want on paper.

Sup­pose you lined up the last dig­its instead. The 8 tenths of $23.8$ would land under the 2 hun­dredths, and you'd be adding 80 paise to 2 paise as if they were the same kind of coin. The answer would come out wrong.

Adding dec­i­mals#

Here are the three steps, 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 never wan­ders. 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, and 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 gen­tler 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, so you're done.

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, and here there are four steps to fol­low 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.

Why does the big­ger num­ber go on top? Well, you can't take 8 rupees away from 3 rupees. Put the big­ger num­ber first and every col­umn has enough to give. And yes, 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's no zero to add this time.

When one num­ber has no point#

A whole num­ber has no point writ­ten down, but you may always 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, and 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 would with whole num­bers. The dec­i­mal point does­n't get in the way at all; you sim­ply 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. A handy approach: gather the num­bers you are adding, gather the num­bers you are tak­ing away, and then do one sin­gle tak­ing away at the end.

Why is that allowed? Think of money com­ing in and money going out of your piggy bank. 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 does­n't 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 first writ­ten as $2476.7740$.

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: $16.25 + 83.75 = 100.00$. Here's a teacher's tip worth keep­ing: 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#

Show 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. $100 - 45.25 = 100.00 - 45.25 = 54.75$