Decimals in weights and money

Dec­i­mals are not only a chap­ter in the text­book. You meet them every day at the kirana shop, on the weigh­ing scale at the veg­etable mar­ket, and on every bill. This les­son shows how one num­ber with a dec­i­mal point can carry two units at once: kilo­grams with grams, and rupees with paise. Once you see why the dig­its sit where they do, chang­ing grams to kilo­grams and paise to rupees becomes a mat­ter of slid­ing dig­its.

Two units in one num­ber

Look at a price label in a shop. It may say ₹$24.50$. That one num­ber holds two things. It holds rupees, and it holds paise.

A dec­i­mal is a num­ber with a dot in it. The dot is called the dec­i­mal point. It splits the whole things on the left from the parts of a thing on the right.

Weights work the same way. A packet can say $2.348$ kg. That is kilo­grams and grams in one num­ber.

What the spots after the point mean

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

That shar­ing has a name. Cut one whole thing into $10$ equal parts. One part is a tenth. Cut it into $100$ equal parts instead. One part is a hun­dredth. Cut it into $1000$ parts and one part is a thou­sandth.

A dec­i­mal is how you write those parts down. The first spot after the point counts tenths. The next spot counts hun­dredths. The next counts thou­sandths. Each spot is ten times smaller than the one before it.

Remem­ber. $1000$ grams $= 1$ kilo­gram. And $100$ paise $=$ ₹$1$.

Both facts work the same way. It always takes the same num­ber of small ones to make one big one. It is $1000$ grams every time, and $100$ paise every time.

So a gram is a thou­sandth of a kilo­gram. A paisa is a hun­dredth of a rupee. Those are parts you can write after a point.

Grams into kilo­grams

Go back to the sugar bag and its $1000$ pinches. One pinch is one gram. So $1$ g is $\displaystyle \frac{1}{1000}$ kg, which is $0.001$ kg.

Look at what hap­pens each time. Every spot to the right is ten times smaller. One step makes a num­ber ten times smaller. Two steps make it a hun­dred times smaller. Three steps make it a thou­sand times smaller.

A gram is a thou­sand times smaller than a kilo­gram. That is three steps. So every digit slides three spots to the right. A digit is one of the num­ber sym­bols, like the $8$ and the $5$ in $85$.

So $85$ g becomes $0.085$ kg. The zero after the point is doing real work. It keeps the tenths spot empty. With­out it you would read $0.85$ kg. That is ten times too heavy.

$5678$ g is more than a kilo­gram. It is $5$ whole kilo­grams and $678$ grams over. 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

Put a ten rupee note on the table. Now imag­ine chang­ing it for the small­est coins there are. You would need $100$ paise to build back just one rupee.

So one paisa is one of a rupee's hun­dred parts. That is ₹$0.01$. A rupee is a hun­dred times big­ger than a paisa. That is two steps. So to change paise into rupees, you divide by $100$.

A shop price label usu­ally shows two spots after the point. A paisa is the small­est coin, so noth­ing smaller needs writ­ing. The first spot counts ten paise coins. The sec­ond counts sin­gle paise. So ₹$0.50$ means fifty paise, or half a rupee.

$225$ paise is more than a rupee. It is $2$ whole rupees and $25$ paise over. So it is ₹$2.25$.

Writ­ing kilo­grams and grams as one num­ber

First turn the grams into kilo­grams. Then add them to the whole kilo­grams. The point keeps the two parts apart.

Write $2$ kg $348$ g in kilo­grams

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

Read the answer back. The $2$ counts kilo­grams. The $348$ after the point counts grams. Noth­ing was lost. The same weight is only writ­ten a new way.

The chart below shows the same idea as a place-value chart. Each box holds one digit, and the point sits between the whole kilo­grams and the parts of a kilo­gram.

More kilo­grams and grams

Watch the empty spots. They must be filled with zeros so that every digit stays in its own col­umn.

Write $3$ kg $5$ g in kilo­grams

$$\begin{aligned}3\text{ kg }5\text{ g} &= 3\text{ kg} + 0.005\text{ kg} \\ &= 3.005\text{ kg}\end{aligned}$$

Five grams is five thou­sandths, so the $5$ goes in the third spot. Two zeros hold the tenths and hun­dredths spots open.

Write $1$ kg $50$ g in kilo­grams

$$\begin{aligned}1\text{ kg }50\text{ g} &= 1\text{ kg} + 0.050\text{ kg} \\ &= 1.05\text{ kg}\end{aligned}$$

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

Going the other way

To turn kilo­grams back into grams, mul­ti­ply by $1000$: every digit slides three spots to the left. So $0.75$ kg is $0.75 \times 1000 = 750$ g.

Writ­ing rupees and paise as one num­ber

Money goes the same way. Turn the paise into rupees. Then add them to the whole rupees.

Write ₹$24$ and $50$ paise in rupees

$$\begin{aligned}&\text{₹}24 \text{ and } 50 \text{ paise} \\ &= \text{₹}24.5\end{aligned}$$

A price label would show ₹$24.50$ instead. It is the same money. A zero between the point and a digit changes the num­ber. That is why $0.085$ kg is not $0.85$ kg. A zero at the very end changes noth­ing.

There is noth­ing after that last zero to push along. So ₹$24.5$ and ₹$24.50$ are the same amount.

Remem­ber. Divide by how­ever many small ones fit in one big one. $1000$ 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 sub­tract them like ordi­nary num­bers. Line up the points first.

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

$$\begin{aligned}1.250 + 2.800 &= 4.050 \\ &= 4.05\text{ kg}\end{aligned}$$

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

A shirt costs ₹$87.50$. You pay with ₹$150$. How much change do you get?

$$\begin{aligned}150.00 - 87.50 &= 62.50\end{aligned}$$

You get ₹$62.50$ back, which is $62$ rupees and $50$ paise.

Prac­tice

Fill in each blank.

Check your answers

Com­mon mis­takes

• Writ­ing $85$ g as $0.85$ kg. Grams need three spots, so it is $0.085$ kg.
• Using three spots for paise. A rupee has $100$ paise, so paise need only two spots: $5$ paise is ₹$0.05$.
• Drop­ping a zero in the mid­dle, such as writ­ing $4$ kg $25$ g as $4.25$ kg.
• Adding dec­i­mals with­out lin­ing up the points.
• Think­ing ₹$24.5$ and ₹$24.50$ are dif­fer­ent. A zero at the very end changes noth­ing.

Key terms

Dec­i­mal point

The dot that sep­a­rates whole units on the left from parts of a unit on the right.

Tenth, hun­dredth, thou­sandth

One of $10$, $100$ or $1000$ equal parts of a whole.

Kilo­gram and gram

$1000$ grams make $1$ kilo­gram.

Rupee and paisa

$100$ paise make ₹$1$.

Digit

One of the num­ber sym­bols $0$ to $9$.

Place value

The worth of a digit because of the spot it sits in.

Answers

1. $250 \text{ g} = 0.25 \text{ kg}$
2. $7 \text{ g} = 0.007 \text{ kg}$
3. $4\text{ kg }25\text{ g} = 4.025 \text{ kg}$
4. $75 \text{ paise} = \text{₹}0.75$
5. $305 \text{ paise} = \text{₹}3.05$
6. $\text{₹}6 \text{ and } 40 \text{ paise} = \text{₹}6.40$