Decimals in weights and money

You meet dec­i­mals every day at the kirana shop, on the weigh­ing scale at the veg­etable mar­ket and on every bill. A packet of atta may say 2.348 kg. A price label may say ₹24.50. Each of these num­bers car­ries two units at once: kilo­grams with grams, or rupees with paise. In this les­son you will see why the dig­its sit where they do, so that chang­ing grams to kilo­grams or paise to rupees becomes easy.

Two facts to remem­ber#

Remem­ber. $1000$ grams make $1$ kilo­gram. $100$ paise make ₹1.

So one gram is one thou­sandth of a kilo­gram. One paisa is one hun­dredth of a rupee. Thou­sandths and hun­dredths are exactly the parts we write after a dec­i­mal point.

Grams into kilo­grams#

Pic­ture a bag of sugar that weighs $1$ kilo­gram. Share it into $1000$ equal pinches. One pinch is one gram.

$$\displaystyle 1 \text{ g} = \frac{1}{1000} \text{ kg} = 0.001 \text{ kg}$$

To change grams into kilo­grams, divide by $1000$. From the ear­lier les­son you know what that means: move the point three places to the left.

Look at $85$ g. It becomes $0.085$ kg. The zero just after the point is doing real work: it keeps the tenths place empty. Leave it out and you would read $0.85$ kg, which is ten times too heavy.

$5678$ g is more than a kilo­gram. It is $5$ whole kilo­grams and $678$ grams more, so it is $5.678$ kg. The part before the point counts whole kilo­grams. The part after it counts the grams left over.

Paise into rupees#

You need $100$ paise to make one rupee. So one paisa is ₹0.01.

To change paise into rupees, divide by $100$: move the point two places to the left.

A price label usu­ally shows two dig­its after the point. The first counts ten-paise parts and the sec­ond counts sin­gle paise. So ₹0.50 means fifty paise, which is half a rupee.

Kilo­grams and grams as one num­ber#

First change the grams into kilo­grams. Then add them to the whole kilo­grams.

Exam­ple 1#

Write $2$ kg $348$ g in kilo­grams.

$$\begin{aligned} 2 \text{ kg } 348 \text{ g} &= 2 \text{ kg} + 348 \text{ g} \\ &= 2 \text{ kg} + 0.348 \text{ kg} \\ &= 2.348 \text{ kg} \end{aligned}$$

The $2$ counts kilo­grams and the $348$ after the point counts grams. Noth­ing is lost. The same weight is sim­ply writ­ten a new way.

Exam­ple 2#

Write $3$ kg $5$ g in kilo­grams.

Five grams is five thou­sandths of a kilo­gram, so the $5$ goes in the third place after the point. Two zeros hold the tenths and hun­dredths places open.

$$3 \text{ kg } 5 \text{ g} = 3 \text{ kg} + 0.005 \text{ kg} = 3.005 \text{ kg}$$

Exam­ple 3#

Write $1$ kg $50$ g in kilo­grams.

$$1 \text{ kg } 50 \text{ g} = 1 \text{ kg} + 0.050 \text{ kg} = 1.05 \text{ kg}$$

The zero at the end of $0.050$ can be dropped. The zero straight after the point can­not.

Going the other way#

To turn kilo­grams back into grams, mul­ti­ply by $1000$: move the point three places to the right.

$$0.75 \text{ kg} = 0.75 \times 1000 \text{ g} = 750 \text{ g}$$

Rupees and paise as one num­ber#

Money works the same way. Change the paise into rupees, then add the whole rupees.

Exam­ple 4#

Write ₹24 and $50$ paise in rupees.

$$\displaystyle 24 \text{ rupees } 50 \text{ paise} = 24 + \frac{50}{100} = 24 + 0.5 = 24.5 \text{ rupees}$$

So it is ₹24.5. A price label would show ₹24.50. That is the same money, because a zero at the very end after the point changes noth­ing.

Exam­ple 5#

Write ₹7 and $8$ paise in rupees.

Eight paise is $\displaystyle \frac{8}{100}$ of a rupee, which is ₹0.08. So the amount is ₹7.08. The zero after the point must stay. With­out it, ₹7.8 would mean 7 rupees and 80 paise.

Going the other way#

To turn rupees into paise, mul­ti­ply by $100$. So ₹12.05 is $12.05 \times 100 = 1205$ paise.

Remem­ber. Divide by how many small units fit in one big unit. $1000$ grams fit in a kilo­gram, so divide grams by $1000$. $100$ paise fit in a rupee, so divide paise by $100$.

Using dec­i­mals in real sums#

Once weights and money are writ­ten as dec­i­mals, you can add and take away like ordi­nary num­bers. Write them with the same num­ber of dig­its after the point and line up the points. The next les­son shows this in detail.

Exam­ple 6#

Amma buys $1$ kg $250$ g of rice and $2$ kg $800$ g of wheat. What is the total weight?

$$1.250 + 2.800 = 4.050$$

The bag holds $4.05$ kg, which is $4$ kg $50$ g.

Exam­ple 7#

A shirt costs ₹87.50. Ravi pays with ₹150. How much change does he get?

$$150.00 - 87.50 = 62.50$$

Ravi gets ₹62.50 back, which is 62 rupees and 50 paise.

Com­mon mis­takes#

• Writ­ing $85$ g as $0.85$ kg. Grams need three places after the point, so it is $0.085$ kg.
• Using three places for paise. A rupee has $100$ paise, so paise need only two places: $5$ paise is ₹0.05.
• Drop­ping a zero in the mid­dle, such as writ­ing $4$ kg $25$ g as $4.25$ kg. It is $4.025$ kg.
• Think­ing ₹24.5 and ₹24.50 are dif­fer­ent. A zero at the very end changes noth­ing.

Prac­tice#

1. Write $250$ g in kilo­grams.
2. Write $7$ g in kilo­grams.
3. Write $4$ kg $25$ g in kilo­grams.
4. Write $2075$ g in kilo­grams.
5. Write $75$ paise in rupees.
6. Write $305$ paise in rupees.
7. Write ₹6 and $40$ paise in rupees.
8. Write $3.6$ kg in grams.
9. Write ₹9.99 in paise.
10. Sita buys $1$ kg $500$ g of toma­toes and $750$ g of onions. What is the total weight in kilo­grams?

Answers#