Adding any number to any number

Adding is some­thing you do every day. You add when you count the sweets in two pack­ets, when you work out how many chil­dren are in two classes, and when a shop­keeper totals your bill of ₹123 and ₹8. Small sums you can do on your fin­gers. Big sums need a plan. This les­son gives you that plan.

Every­thing so far has been build­ing to this. There is no new idea in this les­son: it col­lects the four ways of adding you have already met and shows that they keep work­ing how­ever big the num­bers get.

Think of a num­ber as a heap of beads. Adding two num­bers means push­ing the two heaps together and count­ing how many beads there are now. You can move beads from one heap to the other, or sort them into hun­dreds, tens and units first. The total never changes, because no bead is added and no bead is lost. That is the secret behind all four ways.

The four ways

They all give the same answer, because none of them changes how much there is alto­gether. Choose whichever one makes the sum eas­i­est in your head.

Which way to choose

• If one num­ber is close to a round num­ber, like $198$ or $794$, move an amount across. Make it round first.
• If both num­bers have sev­eral dig­its, split them both. Add hun­dreds, then tens, then units.
• If the sec­ond num­ber is a sin­gle digit, split the sec­ond num­ber so that the first one reaches the next ten.
• If the num­bers are long, write them down the page and add col­umn by col­umn, start­ing with the units.

Worked exam­ples

123 + 2

The units are small, so noth­ing spills over and only the last digit changes.

$123 + 2 = 125$

123 + 8

Split the sec­ond num­ber

$$\begin{aligned}123 + 8 &= 120 + (3 + 8) \\ &= 120 + 11 \\ &= 131\end{aligned}$$

Or move an amount across

$$\begin{aligned}123 + 8 &= 130 + 1 \\ &= 131\end{aligned}$$

Seven of the eight was given to the 123 to make it 130, and one was left over.

The num­ber line below shows those two steps. The first seven small jumps take you from 123 to the round num­ber 130. One more jump lands on 131.

256 + 67

Split them both

$$\begin{aligned}256 + 67 &= 200 + (50 + 60) + (6 + 7) \\ &= 200 + 110 + 13 \\ &= 310 + 13 \\ &= 323\end{aligned}$$

Or move an amount across

$$\begin{aligned}256 + 67 &= 260 + 60 + 3 \\ &= 323\end{aligned}$$

38 + 45

Two two-digit num­bers. Split them both: the tens go together and the units go together.

$$\begin{aligned}38 + 45 &= (30 + 40) + (8 + 5) \\ &= 70 + 13 \\ &= 83\end{aligned}$$

The units made $13$, which is one more ten and $3$ units. That extra ten joins the $70$ to make $80$.

49 + 36

$49$ is just one short of $50$. So move $1$ across from the $36$.

$$\begin{aligned}49 + 36 &= 50 + 35 \\ &= 85\end{aligned}$$

357 + 468

Split them both

$$\begin{aligned}357 + 468 &= (300 + 400) + (50 + 60) + (7 + 8) \\ &= 700 + 110 + 15 \\ &= 810 + 15 \\ &= 825\end{aligned}$$

Or move an amount across

$$\begin{aligned}357 + 468 &= 360 + 465 \\ &= 825\end{aligned}$$

Three was taken from the 468 and given to the 357 to make it the round num­ber 360. The chart below shows the other way, with each place added on its own.

928 + 794

Split them both

$$\begin{aligned}928 + 794 &= (900 + 700) + (20 + 90) + (8 + 4) \\ &= 1600 + 110 + 12 \\ &= 1710 + 12 \\ &= 1722\end{aligned}$$

Or move an amount across

$$\begin{aligned}928 + 794 &= 922 + 800 \\ &= 1722\end{aligned}$$

Six was taken from the 928 and given to the 794, which turned it into a round 800. Adding 800 to any­thing is easy.

1999 + 2345

$1999$ is one short of $2000$, so move $1$ across.

$$\begin{aligned}1999 + 2345 &= 2000 + 2344 \\ &= 4344\end{aligned}$$

4675 + 2398

$2398$ is two short of $2400$. Take $2$ from the $4675$ and give it to the $2398$.

$$\begin{aligned}4675 + 2398 &= 4673 + 2400 \\ &= 6673 + 400 \\ &= 7073\end{aligned}$$

Adding a round num­ber like $2400$ is quick: add the thou­sands, then the hun­dreds.

Big­ger still

The same split­ting works how­ever many dig­its there are. Write them down the page, col­umn by col­umn.

Start with the units col­umn. If a col­umn adds up to ten or more, write the units digit and carry the ten into the next col­umn on the left. For $2358 + 4761$: units $8 + 1 = 9$; tens $5 + 6 = 11$, so write $1$ and carry $1$; hun­dreds $3 + 7 + 1 = 11$, write $1$ and carry $1$; thou­sands $2 + 4 + 1 = 7$. The answer is $7119$.

Dou­bling a num­ber

The dou­ble of a num­ber is what you get by adding the num­ber to itself once.

Dou­bling is use­ful for sums like $7 + 8$. You know the dou­ble $7 + 7 = 14$, and $8$ is one more than $7$, so $7 + 8$ is one more than $14$, which is $15$.

Remem­ber. Mov­ing an amount from one num­ber to the other, or split­ting num­bers into parts, never changes the total.

Your turn

Sin­gle dig­its

A digit added to a two-digit num­ber

Two-digit num­bers together

Com­mon mis­takes

• Writ­ing num­bers down the page with­out lin­ing up the units. Units must sit under units, tens under tens.
• For­get­ting to carry. When a col­umn makes $10$ or more, the extra ten belongs in the next col­umn.
• When mov­ing an amount across, adding to both num­bers. If you give $2$ to one num­ber, you must take $2$ from the other.
• Mix­ing up $7 + 0$ and $7 + 1$. Adding zero leaves a num­ber as it is.
• Stop­ping after adding the parts. In $700 + 110 + 15$ the parts must still be joined to give $825$.

Key terms

Sum

The answer you get when you add num­bers.

Units, tens, hun­dreds

The places a digit can sit in. In $357$ there are $3$ hun­dreds, $5$ tens and $7$ units.

Round num­ber

A num­ber end­ing in zero, such as $130$ or $800$, which is easy to add to.

Carry

To move a ten from one col­umn into the next col­umn on the left.

Dou­ble

A num­ber added to itself once.

Split

To break a num­ber into parts, such as $256 = 200 + 50 + 6$.

Answers

Sin­gle dig­its

1) $0$, 2) $7$, 3) $6$, 4) $8$, 5) $5$, 6) $9$, 7) $8$, 8) $9$, 9) $10$, 10) $10$, 11) $10$, 12) $10$, 13) $10$, 14) $2$, 15) $4$, 16) $6$, 17) $8$, 18) $10$, 19) $12$, 20) $14$, 21) $16$, 22) $18$, 23) $6$, 24) $7$, 25) $8$, 26) $9$, 27) $17$, 28) $12$, 29) $14$, 30) $11$, 31) $15$, 32) $8$, 33) $13$, 34) $13$, 35) $11$, 36) $12$

A digit added to a two-digit num­ber

1) $12$, 2) $18$, 3) $15$, 4) $17$, 5) $27$, 6) $39$, 7) $27$, 8) $31$, 9) $40$, 10) $47$, 11) $45$, 12) $50$, 13) $63$, 14) $81$, 15) $96$, 16) $105$, 17) $91$, 18) $81$

Two-digit num­bers together

1) $20$, 2) $50$, 3) $100$, 4) $100$, 5) $130$, 6) $130$, 7) $100$, 8) $86$, 9) $156$, 10) $78$, 11) $89$, 12) $100$, 13) $85$, 14) $85$, 15) $136$, 16) $83$, 17) $121$, 18) $156$