Grade 3 · 3.MD.A.2

Story problems with grams, kilograms and liters

A sack of seed, a bucket of water, a bottle of shampoo. When a story gives you grams or liters, how do you know whether to add, subtract, multiply or divide?

You need

Liquid volume: liters and millilitersGrade 3 Metric mass unitsGrade 3

Where you are 14 in What measurement is

Measurement12

What measurement is14

▴ 9 above Measurement is approximate: precision of a tool Significant figures

Story problems with grams, kilograms and liters

Word problems with distance, time, volume, mass and money Measurement error

When you reach for it

The two lessons before this one gave names to two amounts. Metric mass units measures how much matter something has, in grams (g) and kilograms (kg). Liquid volume: liters and milliliters measures how much liquid a container holds or how much liquid there is, in milliliters (mL) and liters (L). Once an amount has a number and a unit, it can go into a story: a feed store sells it by the bag, a horse barn pours it into buckets, somebody buys some and somebody uses some up.

This lesson is about those stories, the kind that take one step. Each one gives two amounts and asks for a third, and the step is one of the four operations: add, subtract, multiply or divide. The amounts here always come in the same unit: grams with grams, liters with liters. Nothing has to be changed from one unit to another, and every answer is a whole number of that unit.

You reach for this skill whenever a measured amount is joined, taken away, compared, made of equal copies, or shared out. The arithmetic is the same arithmetic used with counted things. What is new is that the numbers carry units, and the unit has to come along into the answer and make sense there.

The lesson runs over three days. Day 1 adds and subtracts masses and liquid volumes. Day 2 multiplies and divides them. Day 3 reads amounts off a scale picture, compares two amounts, and checks every answer for its unit and its size.

The procedure

Day 1: joining and taking away

A story problem is a small picture in words. Before any number is touched, it helps to ask three questions of it.

What is being measured, and in what unit? Fanta buys two bags of dog treats at the feed store: 365 g of chicken treats and 248 g of beef treats. Both amounts are masses, and both are in grams. That matters. Adding two numbers only makes sense when they count the same thing, and “365 grams” and “248 grams” both count grams. Their sum will count grams too.

What happens in the story? The two bags go into one shopping basket: the amounts are put together. Putting two amounts together and asking for the whole is addition. If instead the story starts with one amount and some of it is poured out, used up or sold, the question is how much is left, which is subtraction.

What is known and what is the question? A tape diagram shows this at a glance. A tape diagram is a bar that stands for an amount, cut into parts, with a bracket for the whole. Draw the known amounts as labeled parts and put a question mark where the unknown goes.

Figure 1Two bags put together. Both parts are known; the whole is the question.

Read the picture. Both parts are known and the whole has the question mark, so the whole is found by adding:

365 + 248 = 613

The answer is 613 g. The unit comes from the parts: grams and grams make grams.

A tape diagram also shows when to subtract. When the bracket over the whole has a number in it and one part has a number, the other part is what is left after that part is taken away. In subtraction the whole and one part are known, and the missing part is whole minus part. The same picture can be read as an addition with a missing part, which is why a subtraction is checked by adding back: the part found plus the part taken must give the whole.

Two more kinds of story use the same two operations. A missing start story tells what was used and what is left, and asks how much there was at first. Both pieces are parts, so the start (the whole) is their sum; Problems with the starting amount missing treats these stories in general. A compare story puts two amounts side by side and asks how much more one is than the other. That gap is the piece of the bigger amount that is left after you match it against the smaller one, so it is found by subtracting, as in Subtraction as the difference. Day 3 draws compare stories with two bars.

The numbers on Day 1 stay within 1{,}000, so the arithmetic is the adding and subtracting of three-digit numbers, with regrouping where it is needed (Regrouping in addition, Regrouping in subtraction).

Day 2: equal amounts and equal shares

Many measured amounts come in equal pieces: bags of the same mass, buckets that each get the same water, doses that are all the same size. A story about equal pieces has three numbers in it: how many pieces, how big each piece is, and the total. Any two of them give the third.

The number of pieces and the size of each are known. Five bags of bird seed have a mass of 4 kg each. The total is 5 equal groups of 4 kg, which is multiplication, as in Equal group problems: 5 \times 4 = 20, so 20 kg. The tape diagram is five equal boxes, each labeled 4 kg, with the question mark on the bracket over them all.

The total and the number of pieces are known. Keita has 45 L of water to share equally among 5 horse buckets. Now the bracket has a number and every box has a question mark. Cutting a whole into a known number of equal shares is division, as in What division means: 45 \div 5 = 9, so each bucket gets 9 L.

The total and the size of each piece are known. A bin holds 36 kg of corn, and it is to be bagged in 4 kg bags. The question is how many bags. Here the division asks “how many 4s fit into 36?”, and the answer is a count of bags, not a mass: 36 \div 4 = 9 bags.

Both kinds of division are undone by multiplication. If 9 L goes into each of 5 buckets, the buckets hold 5 \times 9 = 45 L in all; if 9 bags each hold 4 kg, the bags hold 9 \times 4 = 36 kg. That is how a division is checked.

On Day 2 the multiplying and dividing mostly stay within 100: one-digit numbers of pieces and pieces of one-digit size. The one step past 100 is a one-digit number of pieces that are each a whole number of tens of grams, such as 3 bags of 60 g, which is 3 \times 6 tens, or 18 tens: 180 g.

Day 3: reading pictures and checking the answer

A scale picture. Some stories give their amounts as a picture: a scale’s markings, or the marks on the side of a measuring jug. Read the picture the way Metric mass units and Liquid volume: liters and milliliters showed, by first finding what one space stands for, then use the readings in the story. A difference between two readings on the same scale is the stretch of the scale between them, which is found by subtracting the smaller reading from the larger.

Comparison bars. To compare two amounts, draw one bar for each, lined up at the left, each as long as its amount. The longer bar sticks out past the shorter one, and that piece that sticks out is how much more (heavier, or more water) the bigger amount is.

Checking the unit. Every answer gets a unit, and the unit should be what the question asks about. Grams added to grams give grams. Liters shared among buckets give liters in each bucket. But “how many bags?” or “how many buckets?” is answered with a count of bags or buckets: 36 kg in 4 kg bags is 9 bags, not 9 kg.

Checking the size. Before or after working, ask what size of answer makes sense.

Joining two amounts gives more than either one.
Taking some away leaves less than the start.
Several equal pieces together are more than one piece.
Sharing a whole among two or more leaves each share smaller than the whole.
A rounded estimate, as in Estimating sums and differences, should land near the exact answer.

A check like this catches an answer that came from the wrong operation or a slip in regrouping, because such an answer is usually far from the estimate or on the wrong side of a starting amount.

Worked examples

Example 1

Two bags of dog treats

At the feed store, Fanta buys a bag of chicken dog treats with a mass of 365 g and a bag of beef dog treats with a mass of 248 g. What is the mass of the two bags together?

Solution

Both amounts are masses in grams, so they can be added straight away. The two bags are put together and the question asks for the whole, which is addition (Figure 1).

Add place by place. Ones: 5 + 8 = 13, which is 1 ten and 3 ones; write 3 and carry the ten. Tens: 6 + 4 + 1 = 11 tens, which is 1 hundred and 1 ten; write 1 and carry the hundred. Hundreds: 3 + 2 + 1 = 6. So

365 + 248 = 613

The two bags have a mass of 613 g.

Check the size. Round each mass to the nearest hundred: 365 is about 400 and 248 is about 200, so the total should be about 400 + 200 = 600. The answer 613 g is close to 600, and it is more than either bag alone, as a joined amount must be.

Example 2

Shampoo left in the jug

At the horse barn, a jug holds 900 mL of horse shampoo. Bakary pours 375 mL of it into a bucket of water to wash a horse. How much shampoo is left in the jug?

Solution

Both amounts are liquid volumes in milliliters. The jug starts with the whole, 900 mL, and one part, 375 mL, is poured out. What is left is the other part (Figure 2), so subtract the part from the whole.

Figure 2The whole jug is known and so is the part poured out. The part left is the question.

One way is to count up from 375 to 900. From 375 to 400 is 25. From 400 to 900 is 500. Together that is 25 + 500 = 525. So

900 - 375 = 525

There are 525 mL of shampoo left in the jug.

The same answer comes from regrouping. 900 has no tens and no ones, so rename it as 8 hundreds, 9 tens and 10 ones. Then 10 - 5 = 5 ones, 9 - 7 = 2 tens and 8 - 3 = 5 hundreds.

Check by adding back: the shampoo poured out and the shampoo left must make the whole jug, and 375 + 525 = 900. The answer is also less than 900 mL, as an amount left after pouring must be.

Example 3

Five bags of bird seed

Keita buys 5 bags of bird seed at the feed store. Each bag has a mass of 4 kg. What is the mass of all the bird seed?

Solution

Each bag is the same mass, so the bird seed is 5 equal groups of 4 kg (Figure 3). The total of equal groups is found by multiplying the number of groups by the size of each.

Figure 3Five bags of the same mass. One box for each bag.

5 \times 4 = 20

The bird seed has a mass of 20 kg. Adding the bags one by one gives the same total, 4 + 4 + 4 + 4 + 4 = 20, and the multiplication is the short way to say it.

Check: the unit is kilograms, because each bag was measured in kilograms. The total is more than one bag, as several equal bags together must be.

Example 4

Water for five horse buckets

Keita has 45 L of water. She pours it equally into 5 horse buckets, using all of it. How much water is in each bucket?

Solution

The whole amount, 45 L, is known, and so is the number of equal shares, 5. The size of one share is the question (Figure 4). Cutting a whole into a known number of equal shares is division.

Figure 4The whole is known and so is the number of equal shares. The size of one share is the question.

45 \div 5 = 9

Each bucket gets 9 L of water. The division fact comes from the multiplication fact that undoes it: 5 \times 9 = 45, so five buckets of 9 L each hold the whole 45 L.

Check: the unit is liters, the unit of the water being shared. Each bucket’s 9 L is less than the 45 L that was shared, as one of several equal shares must be.

Example 5

Bagging the corn

A bin at the feed store holds 36 kg of cracked corn. Fanta scoops all of it into bags that hold 4 kg each. How many bags does she fill?

Solution

This time the whole, 36 kg, and the size of each bag, 4 kg, are known. The question is how many bags (Figure 5): how many 4s fit into 36. That is also division.

Figure 5The whole is known and so is the size of one bag. The number of bags is the question.

36 \div 4 = 9

Fanta fills 9 bags. The multiplication fact behind it is 9 \times 4 = 36: nine bags of 4 kg make 36 kg.

Look at the unit of the answer. The numbers in the story were kilograms, but the answer is a count of bags. “9 kg” would be wrong, because the question did not ask for a mass. Compare this with the previous example: there the number of shares was known and the answer was liters in each share; here the size of each share is known and the answer is the number of shares.

Example 6

The hanging scale

At the feed store, customers weigh grain on a hanging scale. Bakary hangs an empty bucket on the scale, and the pointer reads 3 kg. He fills the bucket with oats and hangs it again; now it reads 17 kg. Figure 6 shows the two readings on the scale’s markings. What is the mass of the oats?

Figure 6The scale’s markings, unrolled into a straight line. The oats are the stretch from the empty reading to the full one.

Solution

First read the scale. The numbers 0, 5, 10, 15 and 20 are written on it, and between two numbers there are 5 equal spaces, so each space is 1 kg. The empty bucket’s dot is 3 spaces past 0, at 3 kg, and the full bucket’s dot is 2 spaces past 15, at 17 kg.

The second reading is the bucket and the oats together. The bucket alone is 3 kg of that, so the oats are the rest: the stretch of the scale from 3 to 17. Taking one part away from a whole is subtraction:

17 - 3 = 14

The oats have a mass of 14 kg. Counting spaces on the picture gives the same: 2 spaces from 3 to 5, then 10 spaces from 5 to 15, then 2 more to 17, and 2 + 10 + 2 = 14.

Check: the bucket plus the oats should give the full reading, and 3 + 14 = 17.

Example 7

Hay or straw: which is heavier?

At the horse barn, a bale of hay has a mass of 41 kg and a bale of straw has a mass of 26 kg. How much heavier is the bale of hay than the bale of straw?

Solution

Draw one bar for each bale, lined up at the left (Figure 7). The hay bar is longer. Match the straw bar against it, and the piece of the hay bar that sticks out past the straw is how much heavier the hay is. That piece is what is left of 41 after 26 is taken away, so subtract.

Figure 7Comparison bars. The dashed piece is how much heavier the hay is.

Count up from 26: to 30 is 4, to 40 is 10 more, to 41 is 1 more. In all 4 + 10 + 1 = 15, so

41 - 26 = 15

The hay is 15 kg heavier than the straw.

The word “heavier” sounds like adding more on, but the question is about the gap between two masses that are already given, and a gap is found by subtracting. Check the size and the unit: the gap is smaller than the hay’s 41 kg, it is in kilograms like both bales, and adding it back to the straw gives the hay, 26 + 15 = 41.

Boundaries

The two amounts must be in the same unit. 2 kg and 500 g cannot be added as 2 + 500, because the 2 counts kilograms and the 500 counts grams, and a sum of two counts of different units counts nothing. Every story in this lesson gives its amounts in one unit. Stories that mix units need a change of unit first, which is the work of Word problems with distance, time, volume, mass and money.

Mass and liquid volume are different attributes. A mass says how much matter there is; a liquid volume says how much space a liquid takes up. Grams cannot be added to liters, even when the liters are a liquid with a mass of its own. A story that gives a bucket’s mass and the water’s liters has two separate amounts, not one total.

Whole units only. Every answer here is a whole number of grams, kilograms, milliliters or liters, and every division comes out even. Sharing 38 L equally among 5 buckets does not give a whole number of liters in each, and a share such as “7 L and a part of a liter” needs fractions of a unit, which come later.

A scale reading includes the container. The pointer shows the mass of everything hanging on it. That is why the bucket’s own reading is taken away in the hanging scale example: the reading with the oats in it is not the mass of the oats.

One step. Each story here needs one operation. A story that asks, say, for the water in 8 buckets of 6 L and then how much of a 50 L tank is left takes two steps, as in Problems using several operations.

Where this goes next

Later, the amounts in these stories grow. In Word problems with distance, time, volume, mass and money, a story may give one amount in liters and another in milliliters, so one is changed into the other unit before the operation; it may give a mass as a fraction or a decimal; and it may be about distance, time or money, solved with the same pictures. Stories with two steps join two of today’s operations, one after the other, as in Problems using several operations. The tape diagram and the comparison bars carry on into all of them, and later into the ratio stories of Tape diagrams and sharing in a ratio.

Exercises

Day 1: joining and taking away

At the feed store, a bag of sunflower seeds has a mass of 450 g and a bag of millet has a mass of 385 g. Fanta buys both. What is their total mass?
A water trough at the horse barn holds 280 L when full. Keita has put in 165 L so far. How many more liters does it take to fill the trough?
Bakary pours 236 mL of fly spray into an empty spray bottle, then 158 mL more. How much fly spray is in the bottle?
A sack of horse feed held 50 kg. This week the horses ate 18 kg of it. How much feed is left in the sack?
A box of horse treats has a mass of 940 g. Fanta gives the horses 285 g of the treats. What mass of treats is left in the box?
Keita used 140 mL of hoof oil from a bottle, and 360 mL is left. How much hoof oil was in the bottle at the start? Draw a tape diagram.
At the feed store, a box of dried mealworms has a mass of 340 g and a block of suet has a mass of 520 g. How much heavier is the suet than the mealworms?
A horse drinks 38 L of water on a hot day and 27 L the next day. How many liters does it drink on the two days together?

Day 2: equal amounts and equal shares

Bakary carries 6 bags of chicken feed. Each bag has a mass of 5 kg. What is the mass of all the chicken feed?
Each of 7 horses gets 8 L of water in the morning. How much water is that in all?
72 kg of oats are shared equally among 8 feed bins. How many kilograms go in each bin?
Keita has 42 L of water and fills buckets that hold 6 L each, using all of the water. How many buckets does she fill?
A bag of horse treats has a mass of 70 g. Fanta buys 4 bags. What is the mass of the treats she buys?
Bakary splits 63 kg of hay equally into 9 hay nets. What is the mass of the hay in each net?
Each of 9 horses gets 3 kg of hay at night. How much hay is that in all?
Keita splits 48 mL of vitamin syrup equally into 6 doses. How much syrup is in each dose?

Day 3: pictures, comparisons and checks

A bucket hangs on the feed store scale and the pointer reads 2 kg. Bakary fills it with grain and the scale reads 11 kg. What is the mass of the grain?
Keita’s measuring jug shows 600 mL of water. She pours some into a sprayer, and the jug then shows 250 mL. How much water did she pour?
The bay horse drinks 34 L of water in a day, and the gray pony drinks 18 L. Draw comparison bars. How much more water does the horse drink than the pony?
Bakary has 480 g of sunflower seeds and gives 170 g to a neighbor for her bird feeder. First decide, without working it out exactly, whether more or less than 300 g is left. Then find the exact amount left.
A tub holds 30 L of water. Keita fills 5 L watering cans from it until the tub is empty. How many cans does she fill? What does the number in your answer count?
The corn bin at the feed store had 95 kg in the morning. At closing time it has 37 kg. Draw a tape diagram. How many kilograms of corn were sold during the day?

Reasoning

A bag of seed has a mass of 4 kg and a bag of grit has a mass of 300 g. Why is 304 not the total mass of the two bags in grams or in kilograms?
A whole amount of water is shared equally among 2 or more buckets. Explain why each bucket must get less water than the whole amount.
Write two horse barn stories about liters that are both solved by 32 \div 4: in one, the answer is a number of liters; in the other, it is a number of buckets. Solve both.

Answers

835 g. The two bags are put together, so add: 450 + 385 = 835. Ones 0 + 5 = 5; tens 5 + 8 = 13, write 3 and carry 1 hundred; hundreds 4 + 3 + 1 = 8.
115 L. The full trough is the whole and the 165 L already in is one part, so the missing part is 280 - 165 = 115. Check: 165 + 115 = 280.
394 mL. The two pours go into the same bottle: 236 + 158 = 394. Ones 6 + 8 = 14, write 4 and carry 1 ten; tens 3 + 5 + 1 = 9; hundreds 2 + 1 = 3.
32 kg. Some feed was taken from the whole sack: 50 - 18 = 32. Check: 18 + 32 = 50.
655 g. 940 - 285 = 655. Count up: 285 to 300 is 15, 300 to 900 is 600, 900 to 940 is 40, and 15 + 600 + 40 = 655. Check: 285 + 655 = 940.
500 mL. The tape diagram is one bar with a part of 140 mL (used) and a part of 360 mL (left), and the bracket over the whole holds the question mark. Both parts are known, so the start is 140 + 360 = 500.
180 g. The gap between two masses is found by subtracting: 520 - 340 = 180. Check: 340 + 180 = 520.
65 L. Join the two days: 38 + 27 = 65.
30 kg. 6 equal bags of 5 kg: 6 \times 5 = 30.
56 L. 7 equal amounts of 8 L: 7 \times 8 = 56.
9 kg. The whole and the number of equal shares are known, so divide: 72 \div 8 = 9, because 8 \times 9 = 72.
7 buckets. The whole and the size of each bucket are known, so ask how many 6s make 42: 42 \div 6 = 7, because 7 \times 6 = 42. The answer counts buckets, not liters.
280 g. 4 bags of 70 g: 4 \times 70 = 280, because 4 \times 7 tens is 28 tens.
7 kg. 63 \div 9 = 7, because 9 \times 7 = 63.
27 kg. 9 equal amounts of 3 kg: 9 \times 3 = 27.
8 mL. 48 \div 6 = 8, because 6 \times 8 = 48. Each dose is less than the 48 mL shared.
9 kg. The full reading is the bucket and the grain together, so take the bucket away: 11 - 2 = 9. Check: 2 + 9 = 11.
350 mL. The water poured is the drop from the first reading to the second: 600 - 250 = 350. Check: 250 + 350 = 600.
16 L. The horse’s bar is 34 L long and the pony’s 18 L, lined up at the left; the piece of the horse’s bar that sticks out is the gap. 34 - 18 = 16. Check: 18 + 16 = 34.
More than 300 g; exactly 310 g. Rounded to the nearest hundred, 480 is about 500 and 170 is about 200, and 500 - 200 = 300, so the answer is near 300 g. To see which side of 300 it falls on, add 300 to the amount given away: 170 + 300 = 470, and 470 is less than the 480 g Bakary had, so more than 300 g is left. Exactly: 480 - 170 = 310.
6 cans. How many 5s make 30: 30 \div 5 = 6, because 6 \times 5 = 30. The 6 counts watering cans, not liters; each can holds 5 L.
58 kg. The tape diagram is a bar of 95 kg cut into a part of 37 kg (left) and a part with the question mark (sold). 95 - 37 = 58. Count up: 37 to 40 is 3, 40 to 90 is 50, 90 to 95 is 5, and 3 + 50 + 5 = 58. Check: 37 + 58 = 95.
The 4 counts kilograms and the 300 counts grams. These are units of very different sizes, so 4 + 300 adds two counts of different things, and the result is not a number of grams or a number of kilograms. The two masses can only be added once both are written in one unit, which is the unit changing taught in Word problems with distance, time, volume, mass and money.
The equal shares together make the whole amount. With 2 or more buckets, each share is only one of several pieces of the whole, and the other pieces are not empty, so one share is less than the whole. For example, 45 L shared among 5 buckets gives 9 L each, and 9 is less than 45.
One pair of stories: “Keita pours 32 L of water equally into 4 buckets. How many liters go in each?” The answer is 8 L. “Keita has 32 L of water and fills buckets that hold 4 L each. How many buckets does she fill?” The answer is 8 buckets. Both use 32 \div 4 = 8, because 4 \times 8 = 32. In the first the number of buckets is known and the answer is a volume; in the second the volume of each bucket is known and the answer is a count.