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Grade 4 · 4.MD.A.1 · 5.MD.A.1

Converting metric units by powers of ten

A zipline cable is measured in meters, but the rope for its seat is measured in centimeters. How do you change one metric unit into another, and why is it always a matter of tens?

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Converting metric units by powers of ten

When you reach for it

Burak, Olesya and Urvi are building a zipline in the backyard. They measure the long steel cable with a tape measure, in meters. They measure the rope that holds the seat with a ruler, in centimeters. Indoors, Olesya is making soup. The pot is marked in liters, but her measuring cup is marked in milliliters.

Two measures in different units cannot be compared or put together until they are in the same unit. Is a rope of 480 centimeters longer than 5 meters? Will 5{,}800 milliliters of soup fit in a 6 liter pot? Each question asks you to change one metric unit into another first.

In the metric system that change is always a step of 10, 100 or 1{,}000. So every metric conversion is a multiplication or a division by one of those numbers, and you already know that move: the digits slide over by places, as Multiplying and dividing by ten, one hundred and one thousand shows. This lesson uses that fact to change units in both directions.

The procedure

This lesson takes three days. Day 1 changes a bigger unit into a smaller one. Day 2 goes the other way and works with measures written in two units, such as 2 meters 35 centimeters. Day 3 compares measures given in different units.

Day 1: steps of ten, and multiplying to a smaller unit

The metric system has one basic unit for each kind of measure. Length uses the meter. Mass, which tells how heavy something is, uses the gram. Capacity, which tells how much a container holds, uses the liter. A word in front of the unit, called a prefix, tells how big the unit is compared to the basic one. A prefix means the same thing in front of any unit, as Metric prefixes explains. Three prefixes do almost all the work here:

prefixwhat it meansunits it makes
kilo1{,}000 of the basic unit1 kilometer = 1{,}000 meters, 1 kilogram = 1{,}000 grams
centione hundredth of the basic unit1 meter = 100 centimeters
millione thousandth of the basic unit1 meter = 1{,}000 millimeters, 1 liter = 1{,}000 milliliters

“One hundredth of a meter” means that 100 centimeters laid end to end make one meter, just as 100 cents make one dollar. “One thousandth” means that 1{,}000 of the small unit make one of the basic unit. The short names are \text{km}, \text{m}, \text{cm} and \text{mm} for length, \text{kg} and \text{g} for mass, and \text{L} and \text{mL} for capacity.

There is one more fact between the two small length units. A meter holds 100 centimeters and also 1{,}000 millimeters. Sharing 1{,}000 millimeters among 100 centimeters gives 1{,}000 \div 100 = 10 in each, so

1 \text{ cm} = 10 \text{ mm}.

The figure puts the length units in a row, biggest on the left. Each arrow shows the step between two neighbors.

kilometer meter centimeter millimeter
\times 1{,}000
\div 1{,}000
\times 100
\div 100
\times 10
\div 10
Figure 1Going right, to a smaller unit, you multiply. Going left, to a bigger unit, you divide. The size of the step depends on which two units you are moving between.

Why a bigger unit to a smaller one is a multiplication. Take 3 kilometers. Each kilometer is 1{,}000 meters, so 3 kilometers are 3 groups of 1{,}000 meters:

3 \text{ km} = 3 \times 1{,}000 \text{ m} = 3{,}000 \text{ m}.

The length did not change. It is the same stretch of ground, measured with a smaller ruler, so it takes more of the small unit to cover it. That is why the number gets bigger. In the same way, 4 meters is 4 \times 100 = 400 centimeters, 2 kilograms is 2 \times 1{,}000 = 2{,}000 grams, and 5 liters is 5 \times 1{,}000 = 5{,}000 milliliters.

Multiplying by 100 moves every digit two places to the left and fills the empty places with zeros. Multiplying by 1{,}000 moves every digit three places. So once you know the step, the answer can be written straight down.

A conversion table. When you change many measures between the same two units, a two-column table keeps them in order. The first column holds the bigger unit and the second the smaller one. Every row comes from the same multiplication, so as the first column goes up by 1, the second goes up by the same step each time. The double number line below is the same table drawn as a picture.

0 0 1 100 2 200 3 300 4 400 5 500 meters centimeters
Figure 2The same lengths named in two units. Each meter on the top line lines up with 100 more centimeters on the bottom line.

Day 2: dividing to a bigger unit, and measures in two units

Why a smaller unit to a bigger one is a division. Now the question turns around. How many kilometers are 6{,}000 meters? Each kilometer uses up 1{,}000 meters, so the question is how many groups of 1{,}000 fit into 6{,}000:

6{,}000 \div 1{,}000 = 6, \quad \text{so } 6{,}000 \text{ m} = 6 \text{ km}.

The bigger unit covers more ground, so fewer of them are needed and the number gets smaller. Dividing by 1{,}000 moves every digit three places to the right. You can always check a division by multiplying back: 6 \times 1{,}000 = 6{,}000.

When the groups do not come out even. Not every amount makes whole groups. Take 2{,}350 meters. Two groups of 1{,}000 meters fit, which uses 2{,}000 meters, and 350 meters are left over. The leftover is too short to make another kilometer, so it stays in meters:

2{,}350 \text{ m} = 2 \text{ km } 350 \text{ m}.

A measure written this way, with a bigger unit and a smaller unit side by side, is in mixed units. It is the metric version of 5 feet 4 inches.

From mixed units to one unit. To go back, change the bigger part to the smaller unit and add the parts. For 2 meters 35 centimeters, the 2 meters are 200 centimeters, so

2 \text{ m } 35 \text{ cm} = 200 \text{ cm} + 35 \text{ cm} = 235 \text{ cm}.

The place value chart explains why the digits simply sit side by side here. The 2 meters give 2 hundreds of centimeters, and the 35 centimeters fill the tens and ones. Because of that, a small part needs its zeros. In 5 meters 7 centimeters, the 7 centimeters are 7 ones and the tens place is empty, so the measure is 500 + 7 = 507 centimeters. A zero holds the tens place.

Day 3: comparing measures in different units

Which is longer, 2 kilometers or 1{,}500 meters? The bare numbers say 2 is less than 1{,}500, but the units are not the same, so the bare numbers cannot be compared. Change one measure so that both are in the same unit: 2 kilometers are 2{,}000 meters, and 2{,}000 is more than 1{,}500. So 2 kilometers is the longer distance, by 500 meters.

Either unit would do, but the smaller unit is the better choice. Changing to the smaller unit is a multiplication, so whole numbers stay whole. Once every measure is in the same unit, comparing and ordering them is comparing whole numbers, which you already know how to do.

The same idea works for any number of measures, and for measures in mixed units. Change each one to the smallest unit in the list, then compare. A bigger unit works too when every division comes out even, as in Example 7.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3, 4 and 5 with Day 2, and Examples 6 and 7 with Day 3.

Example 1

The cable and the pot

The zipline cable is 18 meters long. The soup pot holds 4 liters. Write the length of the cable in centimeters and the capacity of the pot in milliliters.

Solution

A meter is 100 centimeters, so 18 meters are 18 groups of 100 centimeters:

18 \times 100 = 1{,}800.

The cable is 1{,}800 centimeters long. The digits 1 and 8 moved two places to the left, and two zeros fill the tens and ones.

A liter is 1{,}000 milliliters, so the pot holds 4 \times 1{,}000 = 4{,}000 milliliters.

Check: both times the new unit is smaller, so the new number should be bigger, and it is.

Example 2

A table for the vegetables

Olesya buys vegetables for the soup by the kilogram, but her recipe lists grams. Make a table that changes 1, 2, 3, 4 and 5 kilograms to grams. Then use it to find how many grams are in 7 kilograms.

Solution

One kilogram is 1{,}000 grams, so each row multiplies the kilograms by 1{,}000.

kilogramsgrams
11{,}000
22{,}000
33{,}000
44{,}000
55{,}000

Each row is 1{,}000 grams more than the row above it, because one more kilogram adds 1{,}000 more grams. Carrying the pattern on, 6 kilograms are 6{,}000 grams and 7 kilograms are 7{,}000 grams. The multiplication says the same thing: 7 \times 1{,}000 = 7{,}000 grams.

Example 3

Back to the bigger unit

Over a weekend of soup making, Urvi uses 12{,}000 milliliters of broth. The zipline platform is 500 centimeters high. Write the broth in liters and the height in meters.

Solution

Each liter uses up 1{,}000 milliliters, so count the groups of 1{,}000 in 12{,}000:

12{,}000 \div 1{,}000 = 12.

Urvi uses 12 liters of broth. Check by multiplying back: 12 \times 1{,}000 = 12{,}000.

Each meter uses up 100 centimeters, so the height is 500 \div 100 = 5 meters. Check: 5 \times 100 = 500.

Both times the new unit is bigger, so the new number is smaller.

Example 4

Meters and centimeters together

Burak measures the rope that will hold the seat. It is 3 meters and 48 centimeters long. Olesya has 2 liters 750 milliliters of soup. Write the rope’s length in centimeters and the soup in milliliters.

Solution

Change the meters to centimeters, then add the extra centimeters. Three meters are 3 \times 100 = 300 centimeters, so

3 \text{ m } 48 \text{ cm} = 300 \text{ cm} + 48 \text{ cm} = 348 \text{ cm}.
100 cm 1 m 100 cm 1 m 100 cm 1 m 48 cm 3 m 48 cm = 348 cm
Figure 3The seat rope in Example 4. Three whole meters are 300 centimeters, and the extra 48 centimeters join them.

The soup works the same way. Two liters are 2 \times 1{,}000 = 2{,}000 milliliters, so

2 \text{ L } 750 \text{ mL} = 2{,}000 \text{ mL} + 750 \text{ mL} = 2{,}750 \text{ mL}.
Example 5

A week of rides

The zipline is 46 meters long. In one week the three friends ride it 30 times in all, each ride the full length. How far do they ride altogether? Give the answer in kilometers and meters.

Solution

Thirty rides of 46 meters are 46 \times 30 meters. Since 46 \times 3 = 138, and multiplying by 10 moves the digits one place left,

46 \times 30 = 1{,}380 \text{ m}.

Now change to kilometers. One group of 1{,}000 meters fits into 1{,}380, with 380 meters left over, which is not enough for a second kilometer. So the friends ride

1{,}380 \text{ m} = 1 \text{ km } 380 \text{ m}.

Check: 1{,}000 + 380 = 1{,}380.

Example 6

Which is more

Olesya has a rope 480 centimeters long. Is it longer or shorter than 5 meters? Is the 3 liter soup jug bigger or smaller than a pot of 3{,}200 milliliters?

Solution

Change the meters to the smaller unit, centimeters: 5 \times 100 = 500 centimeters. Now both lengths are in centimeters, and 480 < 500. So the rope is shorter than 5 meters, by 500 - 480 = 20 centimeters.

Change the liters to milliliters: 3 \times 1{,}000 = 3{,}000 milliliters. Since 3{,}000 < 3{,}200, the jug is smaller than the pot, by 3{,}200 - 3{,}000 = 200 milliliters.

Example 7

Three ropes

Urvi finds three ropes for a second seat. Their labels read 4 meters 6 centimeters, 430 centimeters, and 4{,}100 millimeters. Put them in order from shortest to longest.

Solution

Write all three in one unit. Centimeters work well: two of the labels already use them, and millimeters change to centimeters in one step.

The first rope is 4 meters and 6 centimeters. Four meters are 400 centimeters, and the 6 centimeters fill the ones place, so the rope is 400 + 6 = 406 centimeters.

The second rope is already 430 centimeters.

The third rope is in millimeters, the smaller unit, so divide. Each centimeter is 10 millimeters, and 4{,}100 \div 10 = 410 with nothing left over, so the rope is exactly 410 centimeters.

Now compare whole numbers: 406 < 410 < 430. From shortest to longest the ropes are 4 meters 6 centimeters, then 4{,}100 millimeters, then 430 centimeters.

Check in millimeters instead: the ropes are 4{,}060, 4{,}300 and 4{,}100 millimeters, and 4{,}060 < 4{,}100 < 4{,}300 gives the same order.

Boundaries

  • The steps of ten hold only between metric units. Changing meters to feet, or liters to cups, is not a power of ten, and those conversions use the facts in Customary conversions.
  • Both units must measure the same thing. Meters change to centimeters because both are lengths. Grams cannot change to liters, because one measures how heavy and the other how much space.
  • The size of the step depends on both units, not only on the prefix. Kilometer to meter is 1{,}000, meter to centimeter is 100, and centimeter to millimeter is 10. Kilometer straight to centimeter is the two steps together, 1{,}000 \times 100 = 100{,}000.
  • Dividing to a bigger unit gives a whole number only when the groups come out even. When they do not, the leftover stays in the smaller unit as mixed units, as in Example 5.

Where this goes next

The next step is Word problems with distance, time, volume, mass and money, where conversions like these sit inside longer story problems about distance, mass, capacity, time and money. Later, the leftover part of a measure can be written as a decimal instead of a smaller unit, so 1{,}380 meters becomes 1.38 kilometers; Dividing a decimal by ten, one hundred or one thousand teaches that move, and Multi-step problems with unit conversions uses it. The numbers 10, 100 and 1{,}000 are called powers of ten, because they are made by multiplying tens together, and Exponent notation gives a short way to write them. The same idea of changing units by multiplying returns in Changing the units of a rate.

Exercises

Day 1: bigger unit to smaller unit

  1. Change 7 meters to centimeters.
  2. Change 6 kilometers to meters.
  3. Change 9 liters to milliliters.
  4. Change 8 kilograms to grams.
  5. The tape measure Burak uses is 15 meters long. How many centimeters is that?
  6. Change 12 centimeters to millimeters.
  7. Make a conversion table from liters to milliliters for 1, 2, 3, 6 and 10 liters.
  8. The seat of the zipline hangs from a rope 2 meters long. Olesya says the rope is 20 centimeters. Burak says it is 200 centimeters. Who is right, and why?

Day 2: smaller unit to bigger unit, and mixed units

  1. Change 800 centimeters to meters.
  2. Change 5{,}000 grams to kilograms.
  3. Change 14{,}000 milliliters to liters.
  4. Write 3 meters 72 centimeters in centimeters.
  5. Write 6 liters 250 milliliters in milliliters.
  6. Write 1 kilogram 75 grams in grams.
  7. Write 2{,}430 meters in kilometers and meters.
  8. Urvi pours 3{,}600 milliliters of broth into the pot. Write that in liters and milliliters.

Day 3: comparing

  1. Which is longer, 7 meters or 690 centimeters? By how much?
  2. Which holds more, a 2 liter jug or a bottle of 1{,}950 milliliters? By how much?
  3. Write <, > or = in each blank. (a) 5 kilograms ___ 5{,}000 grams. (b) 3 meters 4 centimeters ___ 340 centimeters. (c) 8{,}000 meters ___ 9 kilometers.
  4. Put these lengths in order from shortest to longest: 250 centimeters, 2 meters 5 centimeters, 2 meters 52 centimeters, 3 meters.
  5. Olesya has 5{,}800 milliliters of soup. Will it all fit in a 6 liter pot? If so, how much room is left?
  6. The zipline cable is 24 meters long. Burak has a safety rope 2{,}350 centimeters long. Is the rope as long as the cable? If not, how much shorter is it?

Reasoning

  1. Changing 9 meters to centimeters gives 900. Explain why the number got bigger even though the length is exactly the same.
  2. Explain why 4 meters 8 centimeters is 408 centimeters and not 48 centimeters.
  3. One kilometer is 1{,}000 meters. Explain how that fact tells you at once how many grams are in a kilogram and how many milliliters are in a liter. Why can you not do the same with feet and pounds?
Answers
  1. 700 centimeters. Each meter is 100 centimeters, so 7 \times 100 = 700.
  2. 6{,}000 meters. Each kilometer is 1{,}000 meters, so 6 \times 1{,}000 = 6{,}000.
  3. 9{,}000 milliliters. Each liter is 1{,}000 milliliters, so 9 \times 1{,}000 = 9{,}000.
  4. 8{,}000 grams. Each kilogram is 1{,}000 grams, so 8 \times 1{,}000 = 8{,}000.
  5. 1{,}500 centimeters. 15 \times 100 = 1{,}500.
  6. 120 millimeters. Each centimeter is 10 millimeters, so 12 \times 10 = 120.
  7. 1 liter is 1{,}000 milliliters, 2 liters are 2{,}000, 3 liters are 3{,}000, 6 liters are 6{,}000, and 10 liters are 10{,}000 milliliters. Each row multiplies the liters by 1{,}000.
  8. Burak is right: the rope is 200 centimeters. Each meter is 100 centimeters, so 2 \times 100 = 200. Twenty centimeters would be only a fifth of one meter.
  9. 8 meters. Each meter uses 100 centimeters, and 800 \div 100 = 8.
  10. 5 kilograms. 5{,}000 \div 1{,}000 = 5.
  11. 14 liters. 14{,}000 \div 1{,}000 = 14; check 14 \times 1{,}000 = 14{,}000.
  12. 372 centimeters. 3 meters are 300 centimeters, and 300 + 72 = 372.
  13. 6{,}250 milliliters. 6 liters are 6{,}000 milliliters, and 6{,}000 + 250 = 6{,}250.
  14. 1{,}075 grams. 1 kilogram is 1{,}000 grams, and 1{,}000 + 75 = 1{,}075. The zero holds the hundreds place.
  15. 2 kilometers 430 meters. Two groups of 1{,}000 meters fit, using 2{,}000, and 2{,}430 - 2{,}000 = 430 meters are left.
  16. 3 liters 600 milliliters. Three groups of 1{,}000 fit, and 3{,}600 - 3{,}000 = 600 milliliters are left.
  17. 7 meters is longer, by 10 centimeters. 7 meters are 700 centimeters, and 700 - 690 = 10.
  18. The 2 liter jug holds more, by 50 milliliters. 2 liters are 2{,}000 milliliters, and 2{,}000 - 1{,}950 = 50.
  19. (a) =, since 5 \times 1{,}000 = 5{,}000. (b) <, since 3 meters 4 centimeters is 304 centimeters and 304 < 340. (c) <, since 9 kilometers is 9{,}000 meters and 8{,}000 < 9{,}000.
  20. 2 meters 5 centimeters, 250 centimeters, 2 meters 52 centimeters, 3 meters. In centimeters they are 205, 250, 252 and 300.
  21. Yes, with 200 milliliters of room left. The pot holds 6 \times 1{,}000 = 6{,}000 milliliters, and 6{,}000 - 5{,}800 = 200.
  22. No, the rope is 50 centimeters shorter. The cable is 24 \times 100 = 2{,}400 centimeters, and 2{,}400 - 2{,}350 = 50.
  23. A centimeter is much smaller than a meter, so it takes many more of them to cover the same length: each meter holds 100 of them, and 9 meters hold 9 \times 100 = 900. The length is the same; only the size of the unit changed, and a smaller unit is counted more times.
  24. The 4 meters are 4 \times 100 = 400 centimeters, and adding 8 more centimeters gives 408. Writing 48 would put the 4 meters in the tens place, as if they were only 40 centimeters. The zero in 408 holds the empty tens place.
  25. The prefix kilo means 1{,}000 of the basic unit, whatever that unit is. So a kilogram is 1{,}000 grams and a kiloliter would be 1{,}000 liters; in the same way milli means one thousandth, so a liter is 1{,}000 milliliters just as a meter is 1{,}000 millimeters. Customary units have no shared prefixes and no single step: a foot is 12 inches, a yard is 3 feet and a pound is 16 ounces, so knowing one fact tells you nothing about another.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0