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. Here you'll see how one num­ber with a dec­i­mal point can carry two units at once: kilo­grams with grams, and rupees with paise. Once you under­stand why the dig­its sit where they do, chang­ing grams to kilo­grams or paise to rupees is just 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 lit­tle num­ber is hold­ing two things at once: rupees and paise.

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

Weights work the same way. A packet of atta can say $2.348$ kg, which is kilo­grams and grams packed into one num­ber.

What the spots after the point mean#

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

This kind of shar­ing has names. Cut one whole thing into $10$ equal parts and one part is a tenth. Cut it into $100$ equal parts instead and one part is a hun­dredth. Cut it into $1000$ parts and one part is a thou­sandth.

A dec­i­mal is sim­ply how you write those parts down. The first spot after the point counts tenths, the next counts hun­dredths, and 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 in the same way: it always takes the same num­ber of small ones to make one big one. It's $1000$ grams every time, and $100$ paise every time.

So a gram is a thou­sandth of a kilo­gram, and a paisa is a hun­dredth of a rupee. Those are exactly the kind of 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.

Notice the pat­tern in the table. Every spot to the right is ten times smaller, so one step makes a num­ber ten times smaller, two steps make it a hun­dred times smaller, and three steps make it a thou­sand times smaller.

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

So $85$ g becomes $0.085$ kg. That zero after the point is doing real work, keep­ing the tenths spot empty. Leave it out and you'd read $0.85$ kg, which is ten times too heavy. Imag­ine the shop­keep­er's face!

$5678$ 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, and the part after it counts the grams left over.

Paise into rupees#

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

So one paisa is one of a rupee's hun­dred parts, which is ₹$0.01$. A rupee is a hun­dred times big­ger than a paisa, which 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 ever needs writ­ing. The first spot counts ten paise coins and 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 neatly 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 to your­self. The $2$ counts kilo­grams and the $348$ after the point counts grams. Noth­ing was lost; the same weight is sim­ply 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 exactly 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, but it's the same money. Here's the dif­fer­ence to keep in mind: a zero between the point and a digit changes the num­ber (that's why $0.085$ kg is not $0.85$ kg), while a zero at the very end changes noth­ing.

There's noth­ing after that last zero for it 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#