Grade 5 · 5.NF.B.6

A fraction of a fraction of an amount

Quang plants part of a berry patch with blueberries and puts netting over part of the blueberries. How much of the whole patch is under netting, and how many pounds is a part of a part of a harvest?

You need

A fraction times a fractionGrade 5 A fraction of a whole numberGrade 4

Unlocks

Two percent changes one after the otherMath 7

Where you are 3 in Fraction of a quantity

Fractions, ratio and percent3

Fractions15

Fraction of a quantity3

A fraction of a whole number Finding the whole when only a fraction of it is known

A fraction of a fraction of an amount

When you reach for it

Quang’s family runs a berry farm. He plants \displaystyle \frac{3}{4} of a berry patch with blueberries. Birds love blueberries, so he puts netting over \displaystyle \frac{2}{3} of the blueberry part. How much of the whole patch is under netting?

That is two cuts in a row. The first cut takes \displaystyle \frac{3}{4} of the patch. The second cut takes \displaystyle \frac{2}{3}, but not of the patch: of what the first cut left, the blueberry part. Finding a part of a part is multiplying a fraction by a fraction, which a fraction times a fraction already explains with area models and a rule. This lesson uses that rule on real amounts: pounds of berries, acres of pasture, gallons of water. It also handles the story where the second cut is of what is left over, which needs one more step first.

The lesson runs over three days. Day 1 finds a fraction of a fraction of one whole thing, with tape diagrams. Day 2 does the same with an amount, such as 60 pounds, and shows two ways that always agree. Day 3 takes stories where the second cut is of “the rest”.

The procedure

Day 1: two cuts in a row

A tape diagram is a long bar that stands for the whole thing. Cutting it into equal parts shows a fraction of it.

Draw the berry patch as a bar and cut it into 4 equal parts. The blueberries are 3 of those parts. Now the second cut works on those 3 parts only. Two thirds of 3 parts is 2 parts, because 3 parts split into 3 equal groups gives 1 part in each group, and 2 groups are 2 parts.

Figure 1The blueberries fill 3 of the 4 parts. Two thirds of those 3 parts is 2 parts, which is \displaystyle \frac{1}{2} of the patch.

The netting covers 2 of the 4 parts of the whole patch. That is \displaystyle \frac{2}{4}, which is \displaystyle \frac{1}{2} of the patch.

The rule for multiplying fractions gives the same answer. “Of” means times, and the fraction that is taken second is written first:

\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}

Here the second cut came out even: 3 parts split into thirds with no new cutting. Often it does not, and then each part is cut again into smaller equal pieces until the second fraction fits. Example 1 shows that.

Two things to keep straight on every problem:

Which whole each fraction is of. The \displaystyle \frac{3}{4} is of the patch. The \displaystyle \frac{2}{3} is of the blueberries. The answer, \displaystyle \frac{1}{2}, is of the patch again. The question always asks about some whole, and the product is a fraction of the first whole.
The second part is smaller than the first. Taking \displaystyle \frac{2}{3} of something leaves less than the something, so the netting, \displaystyle \frac{1}{2}, is less than the blueberries, \displaystyle \frac{3}{4}. A quick look like this catches a slip.

Day 2: a fraction of a fraction of an amount

Now the whole has a size. Udayan picks 60 pounds of raspberries. He sells \displaystyle \frac{3}{4} of them at the farm stand, and \displaystyle \frac{2}{5} of the berries he sells go to a bakery. How many pounds go to the bakery?

Recall how to take a fraction of a whole number (A fraction of a whole number): to find \displaystyle \frac{3}{4} of 60, split 60 into 4 equal groups and take 3 of them.

Way 1: cut the amount, then cut again. Split 60 pounds into 4 equal groups: 60 \div 4 = 15. Three groups are 3 \times 15 = 45 pounds sold. Now split the 45 pounds into 5 equal groups: 45 \div 5 = 9. Two groups are 2 \times 9 = 18 pounds. The bakery gets 18 pounds.

Figure 2Three fourths of 60 pounds is 45 pounds. Two fifths of 45 pounds is 18 pounds.

Way 2: multiply the fractions, then take that fraction once. The bakery gets \displaystyle \frac{2}{5} of \displaystyle \frac{3}{4} of the berries:

\frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}

So the bakery gets \displaystyle \frac{3}{10} of all 60 pounds. Split 60 into 10 equal groups, 60 \div 10 = 6, and take 3 of them: 3 \times 6 = 18 pounds. The same answer.

Why both ways agree. Way 1 multiplies 60 by \displaystyle \frac{3}{4} first and then by \displaystyle \frac{2}{5}. Way 2 multiplies the two fractions first and then multiplies by 60. Written with parentheses for “do this first”:

\begin{aligned} \text{Way 1:}\quad & \frac{2}{5} \times \left(\frac{3}{4} \times 60\right) = \frac{2}{5} \times 45 = 18 \\ \text{Way 2:}\quad & \left(\frac{2}{5} \times \frac{3}{4}\right) \times 60 = \frac{3}{10} \times 60 = 18 \end{aligned}

The same three numbers are multiplied in the same order; only the grouping changes. The associative property of multiplication says the grouping never changes a product: (a \times b) \times c = a \times (b \times c) for any numbers. It holds for fractions just as it does for whole numbers, so the two ways must always give the same answer.

The picture shows it too. In the bottom bar, each of the 5 small parts is 9 pounds, and 9 pounds is \displaystyle \frac{3}{20} of 60, because 60 \div 20 = 3 and 3 \times 3 = 9. Taking 2 of those parts is \displaystyle \frac{6}{20} of 60, the same \displaystyle \frac{6}{20} that Way 2 found.

Which way to use. Either way is correct, so pick the easier one. Way 1 is easy when each cut comes out to a whole number, as here. Way 2 is easier when the first cut does not come out even: \displaystyle \frac{3}{8} of 90 gallons is \displaystyle 33\frac{3}{4} gallons, which is awkward to cut again, while the two fractions may multiply to something simple (Example 4).

Day 3: a fraction of the rest

Many stories cut the leftover part, not the part that was taken. Sooah plants \displaystyle \frac{1}{4} of a berry patch with strawberries. She plants \displaystyle \frac{1}{3} of the rest of the patch with blueberries. What fraction of the patch is blueberries?

The \displaystyle \frac{1}{3} is not of the strawberries and not of the whole patch. It is of the rest, so the rest has to be found first. The whole patch is 1, and the strawberries take \displaystyle \frac{1}{4} of it:

1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}

The rest is \displaystyle \frac{3}{4} of the patch. The blueberries are \displaystyle \frac{1}{3} of that:

\frac{1}{3} \times \frac{3}{4} = \frac{3}{12} = \frac{1}{4}

Figure 3Strawberries take \displaystyle \frac{1}{4} of the patch, so the rest is \displaystyle \frac{3}{4}. One third of the rest is 1 of those 3 parts, \displaystyle \frac{1}{4} of the patch.

The tape diagram shows why. The rest is 3 of the 4 parts, and a third of 3 parts is 1 part, which is \displaystyle \frac{1}{4} of the patch.

The whole patch can now be laid out, and the parts must add back to 1:

Part of the patch	Fraction of the patch
strawberries	\displaystyle \frac{1}{4}
blueberries	\displaystyle \frac{1}{4}
still empty	\displaystyle \frac{1}{2}
all of it	\displaystyle \frac{1}{4} + \frac{1}{4} + \frac{1}{2} = 1

The empty part can be found two ways. It is what is left of the rest after the blueberries: \displaystyle \frac{3}{4} - \frac{1}{4} = \frac{1}{2}. Or it is the other \displaystyle \frac{2}{3} of the rest, since \displaystyle 1 - \frac{1}{3} = \frac{2}{3}: \displaystyle \frac{2}{3} \times \frac{3}{4} = \frac{1}{2}. The words “the rest” can appear more than once in one story, and each time they call for the same move: find what is left, as a fraction of the first whole, before taking a fraction of it.

Notice that “\displaystyle \frac{1}{3} of the rest” and “\displaystyle \frac{1}{3} of the patch” are different amounts. One third of the patch would be \displaystyle \frac{1}{3}; one third of the rest is \displaystyle \frac{1}{4}, which is less, because the rest is less than the patch.

Worked examples

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

Example 1

A third of the clover

On Sooah’s ranch, \displaystyle \frac{2}{5} of a pasture grows clover. Her goats graze \displaystyle \frac{1}{3} of the clover part. What fraction of the pasture do the goats graze?

Solution

Draw the pasture as a bar cut into 5 equal parts; the clover is 2 of them. The second cut asks for a third of those 2 parts, but 2 parts do not split into 3 equal groups. So cut every fifth into 3 smaller equal pieces. The whole bar now has 5 \times 3 = 15 equal pieces, each \displaystyle \frac{1}{15} of the pasture, and the clover is 2 \times 3 = 6 of them.

A third of 6 pieces is 2 pieces, since 6 \div 3 = 2. The goats graze \displaystyle \frac{2}{15} of the pasture. The rule agrees:

\frac{1}{3} \times \frac{2}{5} = \frac{1 \times 2}{3 \times 5} = \frac{2}{15}

Check the size: \displaystyle \frac{2}{15} is less than the clover, \displaystyle \frac{2}{5} = \frac{6}{15}, as a part of a part must be.

Example 2

Straw on the strawberries

Quang grows strawberries on \displaystyle \frac{4}{5} of a field. He spreads straw under \displaystyle \frac{5}{8} of the strawberry plants. What fraction of the field has straw?

Solution

The strawberries are 4 of 5 equal parts of the field. Splitting 4 parts into 8 equal groups needs smaller pieces, so cut each fifth in half. Now the field has 5 \times 2 = 10 equal pieces, each \displaystyle \frac{1}{10} of the field, and the strawberries are 8 of them. That is exactly 8 equal groups of 1 piece. Taking 5 of the groups gives 5 pieces, \displaystyle \frac{5}{10} of the field.

\frac{5}{8} \times \frac{4}{5} = \frac{20}{40} = \frac{1}{2}

Since \displaystyle \frac{5}{10} = \frac{1}{2} too, both agree: straw covers \displaystyle \frac{1}{2} of the field.

Example 3

Lambs on the ranch

Udayan’s ranch has 72 sheep. Of them, \displaystyle \frac{5}{8} are ewes, which are female sheep, and \displaystyle \frac{1}{3} of the ewes have lambs this spring. How many ewes have lambs?

Solution

Cut the amount first. Split 72 into 8 equal groups: 72 \div 8 = 9. Five groups are 5 \times 9 = 45 ewes. A third of 45 is 45 \div 3 = 15.

Multiply the fractions first. The ewes with lambs are \displaystyle \frac{1}{3} of \displaystyle \frac{5}{8} of the sheep:

\frac{1}{3} \times \frac{5}{8} = \frac{5}{24}

Split 72 into 24 equal groups, 72 \div 24 = 3, and take 5 of them: 5 \times 3 = 15.

Both ways give 15 ewes with lambs, as the associative property promises. Here the first way is a little easier, because 72 \div 24 takes more thought than 72 \div 8.

Example 4

When the first cut is not even

A water tank on the ranch holds 90 gallons. Sooah pumps \displaystyle \frac{3}{8} of the water into the horse trough, and the horses drink \displaystyle \frac{4}{9} of the water in the trough. How many gallons do the horses drink?

Solution

Cutting first is possible but awkward. \displaystyle \frac{3}{8} of 90 is \displaystyle \frac{3 \times 90}{8} = \frac{270}{8}, which is \displaystyle 33\frac{3}{4} gallons, and then \displaystyle \frac{4}{9} of \displaystyle 33\frac{3}{4} gallons has to be found.

Multiply the fractions first instead:

\frac{4}{9} \times \frac{3}{8} = \frac{12}{72} = \frac{1}{6}

The horses drink \displaystyle \frac{1}{6} of the tank. Split 90 into 6 equal groups: 90 \div 6 = 15. The horses drink 15 gallons.

The associative property says the awkward way must agree. It does: \displaystyle 33\frac{3}{4} = \frac{135}{4}, and \displaystyle \frac{4}{9} \times \frac{135}{4} = \frac{540}{36} = 15.

Example 5

Pasture from the rest

Sooah’s ranch has 40 acres. Woods cover \displaystyle \frac{1}{5} of the land, and \displaystyle \frac{3}{8} of the rest is pasture. How many acres are pasture?

Solution

The \displaystyle \frac{3}{8} is of the rest, so find the rest first: \displaystyle 1 - \frac{1}{5} = \frac{4}{5} of the land. The pasture is \displaystyle \frac{3}{8} of that:

\frac{3}{8} \times \frac{4}{5} = \frac{12}{40} = \frac{3}{10}

So the pasture is \displaystyle \frac{3}{10} of 40 acres: 40 \div 10 = 4, and 3 \times 4 = 12 acres.

Check by cutting the amount. The woods are \displaystyle \frac{1}{5} of 40, which is 8 acres, so the rest is 40 - 8 = 32 acres. Three eighths of 32 is 32 \div 8 \times 3 = 12 acres.

Example 6

Stand, jam and freezer

Quang picks 84 pounds of blackberries. He sells \displaystyle \frac{1}{3} of them at the stand. He makes \displaystyle \frac{3}{4} of the rest into jam and freezes the berries that are left after that. How many pounds does he freeze, and what fraction of the harvest is that?

Solution

After the stand, the rest is \displaystyle 1 - \frac{1}{3} = \frac{2}{3} of the harvest. The jam takes \displaystyle \frac{3}{4} of that rest, so the freezer gets the rest of the rest, \displaystyle 1 - \frac{3}{4} = \frac{1}{4} of it:

\frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}

Quang freezes \displaystyle \frac{1}{6} of the harvest, and \displaystyle \frac{1}{6} of 84 is 84 \div 6 = 14 pounds.

Check with pounds. The stand takes 84 \div 3 = 28 pounds, leaving 56. Jam takes \displaystyle \frac{3}{4} of 56, which is 56 \div 4 \times 3 = 42 pounds, leaving 56 - 42 = 14 pounds for the freezer. The parts add back to the harvest: 28 + 42 + 14 = 84.

Example 7

Three cuts of strawberries

The berry farm picks 96 quarts of strawberries. On Friday it sells \displaystyle \frac{3}{8} of them. On Saturday it sells \displaystyle \frac{4}{5} of the rest. Of what is still left, \displaystyle \frac{1}{2} goes to the food bank. What fraction of the strawberries goes to the food bank, and how many quarts is that?

Solution

Follow the strawberries one cut at a time, keeping every fraction as a fraction of all 96 quarts.

After Friday, the rest is \displaystyle 1 - \frac{3}{8} = \frac{5}{8} of the strawberries.

Saturday sells \displaystyle \frac{4}{5} of that rest, so what is still left is the other \displaystyle 1 - \frac{4}{5} = \frac{1}{5} of it:

\frac{1}{5} \times \frac{5}{8} = \frac{5}{40} = \frac{1}{8}

The food bank gets \displaystyle \frac{1}{2} of that:

\frac{1}{2} \times \frac{1}{8} = \frac{1}{16}

So \displaystyle \frac{1}{16} of the strawberries goes to the food bank, and 96 \div 16 = 6 quarts.

Check with quarts. Friday sells 96 \div 8 \times 3 = 36, leaving 60. Saturday sells 60 \div 5 \times 4 = 48, leaving 12. Half of 12 is 6. The food bank’s \displaystyle \frac{1}{16} is a fraction of a fraction of a fraction: three cuts in a row, and the associative property lets them be multiplied in one go, \displaystyle \frac{1}{2} \times \frac{1}{5} \times \frac{5}{8} = \frac{1}{16}.

Boundaries

The second fraction must be of the first part. Multiplying is right only when the second cut works on what the first cut took or left. If both fractions are of the same whole (“\displaystyle \frac{1}{4} of the field is corn and \displaystyle \frac{1}{3} of the field is oats”), the parts sit side by side, and they are added or compared, not multiplied.
“The rest” needs a subtraction first. The fraction of the rest is 1 minus the first fraction, and only then is the second fraction taken of it. The fractions are of one whole, so 1 means that whole.
Equal parts mean equal amounts. “\displaystyle \frac{2}{3} of the blueberries” means \displaystyle \frac{2}{3} of the amount, the area or the weight, not 2 of 3 rows that might be different sizes.
Counted things must come out whole. For pounds or gallons, an answer like \displaystyle 3\frac{3}{4} pounds is fine. For sheep or horses, a fraction of a fraction must give a whole number, or the story does not fit its numbers.
Fractions less than 1 make the part smaller. Every fraction in this lesson is less than 1, so each cut gives less than before. A fraction greater than 1, such as \displaystyle \frac{3}{2} of an amount, gives more; Multiplying as scaling and Multiplying mixed numbers take that up.

Where this goes next

Next in this unit, Multiplying as scaling tells how big a product will be before working it out, by comparing each factor with 1, and Multiplying mixed numbers carries the same “of” to amounts like \displaystyle 2\frac{1}{2} times a number. Running the cuts backward, from a known part to the whole it came from, is Finding the whole when only a fraction of it is known. Much later, two percent changes in a row, such as a discount and then a second discount on the new price, work exactly like two cuts in a row: each change applies to what the last one left, so the multipliers are multiplied (Two percent changes one after the other). Sharing whole things into equal parts, the other face of a fraction, is A fraction as division.

Exercises

Exercises 1 to 7 go with Day 1, 8 to 15 with Day 2, 16 to 22 with Day 3, and the reasoning group draws on all three days. Give fractions in simplest form.

Day 1: two cuts in a row

Quang plants \displaystyle \frac{2}{3} of a patch with raspberries and covers \displaystyle \frac{1}{2} of the raspberry part with netting. What fraction of the patch has netting?
On the ranch, \displaystyle \frac{4}{5} of a field is hay. Udayan cuts \displaystyle \frac{1}{4} of the hay part on Monday. What fraction of the field does he cut on Monday?
A row is \displaystyle \frac{3}{8} blackberry bushes. Sooah prunes \displaystyle \frac{2}{3} of the blackberry part. What fraction of the row does she prune? Use a tape diagram.
Sooah waters \displaystyle \frac{5}{6} of a patch, and \displaystyle \frac{1}{2} of the watered part gets fertilizer too. What fraction of the patch gets fertilizer?
Of the berry farm’s land, \displaystyle \frac{5}{9} is in berries, and \displaystyle \frac{3}{5} of the berry land is blueberries. What fraction of the farm’s land is blueberries?
Sooah fences \displaystyle \frac{1}{4} of a pasture, and new grass grows on \displaystyle \frac{2}{3} of the fenced part. What fraction of the pasture has new grass? Say how the tape diagram has to be cut again.
Describe a tape diagram for \displaystyle \frac{3}{4} of \displaystyle \frac{2}{7} of a patch: how many parts the bar starts with, how it is cut again, and how many pieces are taken. Give the answer.

Day 2: a fraction of a fraction of an amount

Quang has 80 pounds of feed. He gives \displaystyle \frac{3}{4} of it to the cattle, and \displaystyle \frac{1}{3} of the cattle’s feed goes to the calves. How many pounds go to the calves? Find it both ways.
A shed holds 150 pounds of oats. Udayan uses \displaystyle \frac{2}{5} of the oats in March, and \displaystyle \frac{5}{6} of the March oats go to the horses. How many pounds do the horses get?
Of the 64 cattle on the ranch, \displaystyle \frac{3}{4} graze the north pasture, and \displaystyle \frac{1}{6} of those are calves. How many calves graze the north pasture?
The farm stand has 120 jars of jam. Of them, \displaystyle \frac{3}{5} are blueberry, and \displaystyle \frac{2}{9} of the blueberry jars are small. How many small blueberry jars are there?
A cooler holds 20 gallons of lemonade. Sooah pours \displaystyle \frac{5}{6} of it into pitchers, and \displaystyle \frac{3}{5} of the lemonade in pitchers is drunk at lunch. How many gallons are drunk at lunch? Try cutting first, then multiplying the fractions first, and say which is easier.
Udayan has 10 pounds of blueberries. He puts \displaystyle \frac{3}{4} of them in baskets and gives \displaystyle \frac{1}{2} of the basket berries to his neighbor. How many pounds does the neighbor get?
The farm stand takes in 240 dollars on Sunday. Berries bring in \displaystyle \frac{5}{8} of the money, and raspberries bring in \displaystyle \frac{2}{5} of the berry money. How many dollars do raspberries bring in?
Quang finds \displaystyle \frac{1}{2} of \displaystyle \frac{4}{5} of 70 pounds. Sooah finds \displaystyle \frac{4}{5} of \displaystyle \frac{1}{2} of 70 pounds. Find both, and explain why they match.

Day 3: a fraction of the rest

Udayan plants \displaystyle \frac{1}{3} of a patch with strawberries and \displaystyle \frac{1}{2} of the rest with blueberries. What fraction of the patch is blueberries, and what fraction is still empty?
A ranch has 48 acres. Woods cover \displaystyle \frac{1}{6} of it, and \displaystyle \frac{3}{5} of the rest is pasture. How many acres are pasture?
Sooah picks 100 pounds of blueberries. She sells \displaystyle \frac{3}{10} of them on Friday and \displaystyle \frac{4}{7} of the rest on Saturday. How many pounds does she sell on Saturday, and how many are left?
The ranch has 36 horses. Of them, \displaystyle \frac{1}{4} are ponies, and \displaystyle \frac{1}{3} of the rest are mares. How many mares are there, and what fraction of the horses is that?
A bag holds 30 pounds of chicken feed. Quang uses \displaystyle \frac{2}{5} of it on Monday and \displaystyle \frac{1}{2} of what is left on Tuesday. How many pounds are left after Tuesday, and what fraction of the bag is that?
The farm picks 144 quarts of raspberries. It sends \displaystyle \frac{1}{3} of them to a bakery, sells \displaystyle \frac{5}{8} of the rest at the stand, and freezes \displaystyle \frac{1}{2} of what is left after that. How many quarts are frozen, and what fraction of the harvest is that?
A field has 24 acres. In Sooah’s plan, \displaystyle \frac{1}{4} of the field is corn and \displaystyle \frac{1}{3} of the field is oats. In Quang’s plan, \displaystyle \frac{1}{4} of the field is corn and \displaystyle \frac{1}{3} of the rest is oats. How many acres of oats does each plan have?

Reasoning

Quang always cuts the amount first. Udayan always multiplies the fractions first. Explain why they always get the same answer, naming the property, and say when each way is easier.
The stand sells \displaystyle \frac{1}{2} of its berries in the morning and \displaystyle \frac{1}{2} of the rest in the afternoon. Is anything left? What fraction? Explain why two halves did not sell everything.
Explain why, when both fractions are less than 1, a fraction of a fraction of an amount is always less than the part from the first cut alone.

Answers

\displaystyle \frac{1}{3} of the patch. The raspberries are 2 of 3 parts, and half of 2 parts is 1 part; \displaystyle \frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}.
\displaystyle \frac{1}{5} of the field. The hay is 4 of 5 parts, a fourth of them is 1 part; \displaystyle \frac{1}{4} \times \frac{4}{5} = \frac{4}{20} = \frac{1}{5}.
\displaystyle \frac{1}{4} of the row. The bar has 8 parts, 3 of them blackberries; two thirds of 3 parts is 2 parts, \displaystyle \frac{2}{8} = \frac{1}{4}. The rule: \displaystyle \frac{2}{3} \times \frac{3}{8} = \frac{6}{24} = \frac{1}{4}.
\displaystyle \frac{5}{12} of the patch. Cut each sixth in half to get 12 pieces; the watered part is 10 of them, and half is 5. \displaystyle \frac{1}{2} \times \frac{5}{6} = \frac{5}{12}.
\displaystyle \frac{1}{3} of the land. \displaystyle \frac{3}{5} \times \frac{5}{9} = \frac{15}{45} = \frac{1}{3}.
\displaystyle \frac{1}{6} of the pasture. The fenced part is 1 of 4 parts and cannot split into thirds, so cut every fourth into 3, making 12 pieces; the fenced part is 3 of them and two thirds of it is 2 pieces, \displaystyle \frac{2}{12} = \frac{1}{6}.
\displaystyle \frac{3}{14} of the patch. Start with 7 parts and take 2. To split 2 parts into 4 equal groups, cut each seventh in half: 14 pieces, 4 of them in the patch’s part, one per group. Take 3 groups, 3 pieces: \displaystyle \frac{3}{4} \times \frac{2}{7} = \frac{6}{28} = \frac{3}{14}.
20 pounds. Cut first: 80 \div 4 \times 3 = 60, and 60 \div 3 = 20. Fractions first: \displaystyle \frac{1}{3} \times \frac{3}{4} = \frac{1}{4}, and 80 \div 4 = 20.
50 pounds. \displaystyle \frac{2}{5} of 150 is 60, and \displaystyle \frac{5}{6} of 60 is 50. Or \displaystyle \frac{5}{6} \times \frac{2}{5} = \frac{10}{30} = \frac{1}{3}, and 150 \div 3 = 50.
8 calves. \displaystyle \frac{3}{4} of 64 is 48, and 48 \div 6 = 8. Or \displaystyle \frac{1}{6} \times \frac{3}{4} = \frac{1}{8}, and 64 \div 8 = 8.
16 jars. \displaystyle \frac{3}{5} of 120 is 72, and \displaystyle \frac{2}{9} of 72 is 16. Or \displaystyle \frac{2}{9} \times \frac{3}{5} = \frac{6}{45} = \frac{2}{15}, and 120 \div 15 \times 2 = 16.
10 gallons. Cutting first, \displaystyle \frac{5}{6} of 20 is \displaystyle 16\frac{2}{3} gallons, which is hard to take \displaystyle \frac{3}{5} of. Multiplying first is easier: \displaystyle \frac{3}{5} \times \frac{5}{6} = \frac{15}{30} = \frac{1}{2}, and half of 20 is 10.
\displaystyle 3\frac{3}{4} pounds. \displaystyle \frac{1}{2} \times \frac{3}{4} = \frac{3}{8}, and \displaystyle \frac{3}{8} of 10 is \displaystyle \frac{30}{8} = 3\frac{6}{8} = 3\frac{3}{4}.
60 dollars. \displaystyle \frac{5}{8} of 240 is 150, and \displaystyle \frac{2}{5} of 150 is 60. Or \displaystyle \frac{2}{5} \times \frac{5}{8} = \frac{1}{4}, and 240 \div 4 = 60.
Both are 28 pounds. Quang: \displaystyle \frac{4}{5} of 70 is 56, half is 28. Sooah: half of 70 is 35, and \displaystyle \frac{4}{5} of 35 is 28. The associative property lets each of them multiply the two fractions first, and the commutative property says the order of the two fractions does not matter, so both take \displaystyle \frac{1}{2} \times \frac{4}{5} = \frac{4}{5} \times \frac{1}{2} = \frac{2}{5} of 70.
Blueberries \displaystyle \frac{1}{3}, empty \displaystyle \frac{1}{3}. The rest is \displaystyle 1 - \frac{1}{3} = \frac{2}{3}, and \displaystyle \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}. Empty is the other half of the rest, also \displaystyle \frac{1}{3}; and \displaystyle \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1.
24 acres. The rest is \displaystyle \frac{5}{6}; \displaystyle \frac{3}{5} \times \frac{5}{6} = \frac{1}{2}, and half of 48 is 24. Check: woods 8 acres, rest 40, and \displaystyle \frac{3}{5} of 40 is 24.
40 pounds on Saturday, 30 pounds left. Friday sells 30 pounds, leaving 70; \displaystyle \frac{4}{7} of 70 is 40, and 70 - 40 = 30.
9 mares, \displaystyle \frac{1}{4} of the horses. Ponies are 9, the rest 27, and a third of 27 is 9. As fractions, \displaystyle \frac{1}{3} \times \frac{3}{4} = \frac{1}{4}.
9 pounds, \displaystyle \frac{3}{10} of the bag. Monday uses 12 pounds, leaving 18; Tuesday uses 9, leaving 9. As fractions, \displaystyle \frac{1}{2} \times \frac{3}{5} = \frac{3}{10}, and \displaystyle \frac{3}{10} of 30 is 9.
18 quarts, \displaystyle \frac{1}{8} of the harvest. After the bakery, \displaystyle \frac{2}{3} is left (96 quarts); the stand sells \displaystyle \frac{5}{8} of that, leaving \displaystyle \frac{3}{8} \times \frac{2}{3} = \frac{1}{4} (36 quarts); half of that is \displaystyle \frac{1}{8}, and 144 \div 8 = 18.
Sooah’s plan has 8 acres of oats, Quang’s has 6. Sooah: \displaystyle \frac{1}{3} of 24 is 8. Quang: the rest is 18 acres, and \displaystyle \frac{1}{3} of 18 is 6.
Cutting first works out \displaystyle \frac{a}{b} \times \left(\frac{c}{d} \times N\right); multiplying first works out \displaystyle \left(\frac{a}{b} \times \frac{c}{d}\right) \times N. The numbers and their order are the same and only the grouping differs, and the associative property of multiplication says grouping never changes a product. Cutting first is easier when each cut gives a whole number; multiplying first is easier when the first cut gives a fraction or when the two fractions simplify to something simple.
Yes, \displaystyle \frac{1}{4} is left. The afternoon sells \displaystyle \frac{1}{2} of the rest, \displaystyle \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}, so in all \displaystyle \frac{1}{2} + \frac{1}{4} = \frac{3}{4} is sold. The second half is of the rest, which is only half of the berries, so it sells a quarter of the berries, not a half.
The first cut gives some part. The second cut takes a fraction less than 1 of that part, which means splitting it into equal pieces and taking fewer pieces than there are. Some of the part is always left out, so what is taken is less than the part. On a tape diagram the second shading always sits inside the first and never fills it.