Metric Conversions with Powers of Ten and Mixed Units | Grade 5 Math | LevelKite

You already know metric facts like $1 \text{ m} = 100 \text{ cm}$ and $1 \text{ kg} = 1{,}000 \text{ g}$ . In this lesson you will use them to convert quickly, to add and subtract measurements, and to solve problems that take more than one step.

To convert a measurement is to write the same amount in a different unit. Think about one stick. It is $1$ m long, and it is also $100$ cm long. The stick did not grow. A centimeter is smaller than a meter, so it takes more centimeters to match the same length.

That tells you which way to go:

To a smaller unit: you need more of them, so you multiply.
To a larger unit: you need fewer of them, so you divide.

What do you multiply or divide by? Look at the metric facts you know. The numbers $10$ , $100$ and $1{,}000$ keep showing up. Each one is a power of ten. That is the name for $10$ and for any number you make by multiplying tens together.

$10$ is one ten.
$100$ is two tens multiplied together: $10 \times 10$ .
$1{,}000$ is three tens multiplied together: $10 \times 10 \times 10$ .

You can write a power of ten in a short way with an exponent. In a power of ten, the exponent is the small raised number that counts how many tens are multiplied. So $10^3$ means $10 \times 10 \times 10$ , which is $1{,}000$ .

Metric facts you know, sorted by their power of ten

Power of ten	Tens multiplied	With an exponent	Metric facts that use it
$10$	$10$ (one ten)	$10^1$	$1 \text{ cm} = 10 \text{ mm}$
$100$	$10 \times 10$	$10^2$	$1 \text{ m} = 100 \text{ cm}$
$1{,}000$	$10 \times 10 \times 10$	$10^3$	$1 \text{ km} = 1{,}000 \text{ m}$ , $1 \text{ kg} = 1{,}000 \text{ g}$ , $1 \text{ L} = 1{,}000 \text{ mL}$

Multiplying or dividing by a power of ten is quick because of place value. In any number, each place is worth $10$ times the place on its right. A ten is worth $10$ ones, and a one is worth $10$ tenths.

So when you multiply a number by $10$ , every digit shifts, or moves, one place to the left. When you divide by $10$ , every digit moves one place to the right. The digits keep their order.

A bigger power of ten has more tens multiplied together, so the digits move more places. The exponent tells you how many: one place for $10^1$ , two places for $10^2$ and three places for $10^3$ .

The next table shows this for two lengths.

Each digit moves two places when you multiply or divide by 100. A dash means that place has no digit.

Measurement	Hundreds	Tens	Ones	Tenths	Hundredths
$3.6$ m	–	–	$3$	$6$	–
$3.6 \times 100 = 360$ cm	$3$	$6$	$0$	–	–
$45$ cm	–	$4$	$5$	–	–
$45 \div 100 = 0.45$ m	–	–	$0$	$4$	$5$

Read the first two rows. A length of $3.6$ m is the same as $360$ cm. A centimeter is the smaller unit, so you multiply by $100$ . The $3$ moves from the ones place to the hundreds place. The $6$ moves from the tenths place to the tens place. Both digits moved two places to the left. The ones place is now empty, so a $0$ goes there.

Read the last two rows. A length of $45$ cm is the same as $0.45$ m. A meter is the larger unit, so you divide by $100$ . The $4$ moves from the tens place to the tenths place. The $5$ moves from the ones place to the hundredths place. Both digits moved two places to the right. A $0$ goes in the ones place to show that there are no whole meters.

Each time, the digits kept their order. Only their places changed.

Key concept

Convert with a power of ten, then work in one unit

Powers of ten link metric units of the same kind, such as two units of length. So converting is a shift of the digits.

Larger unit to smaller unit: multiply. The digits move left.
Smaller unit to larger unit: divide. The digits move right.
The exponent gives the number of places: one for $10^1$ , two for $10^2$ , three for $10^3$ .

Use this to write every measurement in one unit before you add or subtract, because you can only add or subtract numbers that count the same unit.

Check: for any amount greater than zero, a smaller unit gives a bigger number, and a larger unit gives a smaller number.

Worked example

Convert to a larger unit

Write $85$ mL in liters.

Find the fact that links the two units: 1 L is 1,000 mL. A liter is the larger unit, so you need fewer liters than milliliters. That means you divide by 1,000.
$85 \div 1{,}000$
Count the places. 1,000 is three tens multiplied together, so each digit moves three places to the right. The 8 moves from the tens place to the hundredths place. The 5 moves from the ones place to the thousandths place.
$1{,}000 = 10 \times 10 \times 10 = 10^3$
Fill the empty places. No digit landed in the ones place or the tenths place, so write a 0 in each of them. The zeros keep the 8 and the 5 in their new places.
$85 \div 1{,}000 = 0.085, \quad \text{so } 85 \text{ mL} = 0.085 \text{ L}$
Check by going back. Multiplying by 1,000 moves each digit three places to the left, so you should land on the number you started with.
$0.085 \times 1{,}000 = 85$

Answer

Written in liters, $85$ mL is $0.085$ L.

A measurement in mixed units uses two units together, like $3$ m $5$ cm. It means $3$ whole meters and $5$ more centimeters.

Think about adding $2$ m and $30$ cm. The answer is not $32$ of anything. The $2$ counts meters and the $30$ counts centimeters, and those units are different sizes. You can add or subtract the numbers only when they count the same unit.

So first write every measurement in one unit. You may pick either unit. Both choices give the same length, because converting never changes the amount.

In centimeters: $2$ m is $200$ cm, and $200 + 30 = 230$ . The total is $230$ cm.
In meters: $30$ cm is $0.3$ m, and $2 + 0.3 = 2.3$ . The total is $2.3$ m.

These are the same length, because $2.3 \times 100 = 230$ .

Worked example

Add and subtract mixed units

One rope is $3$ m $5$ cm long. Another rope is $1$ m $48$ cm long. How long are the two ropes together? How much longer is the first rope than the second?

Write the first length in one unit. Centimeters work well here because both lengths become whole numbers. A centimeter is the smaller unit, so multiply the meters by 100: 3 m is 3 × 100 = 300 cm. Then add on the 5 cm.
$3 \text{ m } 5 \text{ cm} = 300 \text{ cm} + 5 \text{ cm} = 305 \text{ cm}$
Do the same for the second rope. 1 m is 1 × 100 = 100 cm. Then add on the 48 cm.
$1 \text{ m } 48 \text{ cm} = 100 \text{ cm} + 48 \text{ cm} = 148 \text{ cm}$
Add to find the length of the two ropes together. Both numbers count centimeters now, so you can add them.
$305 \text{ cm} + 148 \text{ cm} = 453 \text{ cm}$
Subtract to find how much longer the first rope is. In one unit this is a simple subtraction. You do not have to take 48 cm away from only 5 cm.
$305 \text{ cm} - 148 \text{ cm} = 157 \text{ cm}$
Write each result in mixed units, like the lengths in the problem. Every 100 cm makes 1 m. So 453 cm is 400 cm and 53 cm, which is 4 m 53 cm. In the same way, 157 cm is 100 cm and 57 cm, which is 1 m 57 cm.
$453 \text{ cm} = 4 \text{ m } 53 \text{ cm} \qquad 157 \text{ cm} = 1 \text{ m } 57 \text{ cm}$

Answer

Together the ropes are $453$ cm long, which is $4$ m $53$ cm. The first rope is $157$ cm longer than the second, which is $1$ m $57$ cm.

A multi-step problem needs more than one calculation to reach the answer. When its measurements are in different units, at least one of the steps is a conversion.

This plan helps:

Read what the question asks for, and in which unit.
Pick one unit to work in. Convert any measurement that is not in that unit.
Do the other calculations.
If the question asks for a different unit, convert your result at the end.

The conversion can come before the other steps, after them, or both. What matters is that you never add or subtract numbers that count different units.

Worked example

A problem with more than one step

You have a $2.5$ kg bag of flour. Each loaf of bread uses $350$ g of flour. You bake $6$ loaves. How many kilograms of flour are left?

The question asks for kilograms, but the loaves are measured in grams. Work in grams first, because they keep the numbers whole here. Each of the 6 loaves uses 350 g, so multiply to find the flour you used.
$6 \times 350 \text{ g} = 2{,}100 \text{ g}$
Write the bag in grams too, so you can subtract. A gram is the smaller unit, so multiply by 1,000. The digits move three places to the left, and zeros fill the empty tens and ones places.
$2.5 \times 1{,}000 = 2{,}500, \quad \text{so } 2.5 \text{ kg} = 2{,}500 \text{ g}$
Subtract the flour you used from the flour you started with. Both numbers count grams now.
$2{,}500 \text{ g} - 2{,}100 \text{ g} = 400 \text{ g}$
Convert the result to kilograms, because that is the unit the question asks for. A kilogram is the larger unit, so divide by 1,000. The digits move three places to the right, so the 4 moves from the hundreds place to the tenths place.
$400 \div 1{,}000 = 0.4, \quad \text{so } 400 \text{ g} = 0.4 \text{ kg}$

Answer

You have $0.4$ kg of flour left. That is the same amount as $400$ g.

Here is the big idea to keep. Powers of ten link metric units of the same kind, so converting is a shift of the digits. Multiply to reach a smaller unit, and divide to reach a larger unit. Before you add or subtract, write every measurement in one unit. In a problem with more than one step, convert so that all the numbers count the same unit, and give your answer in the unit the question asks for.