Why flipping the second fraction works
A scoop holds three quarters of a cup. How many scoops are in a bag of flour, and why does turning the fraction upside down and multiplying give the answer?
Where you are 6 in Dividing fractions
Why this exists
Dividing whole numbers has a plain meaning. 12 \div 3 asks how many threes fit in twelve, and the answer is 4. The same question still makes sense when the numbers are fractions: in a kitchen, a cook with 3 cups of flour and a measuring cup that holds \displaystyle \frac{3}{4} of a cup can ask how many scoops the flour makes. That meaning is the subject of dividing fractions: how many fit, and the sharing meaning is in dividing fractions: sharing into equal parts.
Counting scoops on a picture works, but it is slow, and a picture cannot easily show \displaystyle \frac{7}{8} \div \frac{2}{5}. There is a short rule: turn the second fraction upside down and multiply. A rule handed over without a reason is easy to misremember (which fraction flips?) and impossible to trust. This lesson shows what dividing by a fraction means, counts the answer directly, and then proves three different ways that the rule must give the same answer.
Definition
Division. For any number a and any number b that is not 0, a \div b is the number that b must be multiplied by to give a:
This is the same link between multiplication and division that holds for whole numbers (multiplication and division undo each other). In the kitchen it says: a \div b is the number of b-cup scoops in a cups, and a part of a scoop counts as that fraction of a scoop.
Reciprocal. The reciprocal of a fraction \displaystyle \frac{c}{d}, where c and d are whole numbers other than 0, is \displaystyle \frac{d}{c}: the same fraction turned upside down. The reciprocal of a whole number n is \displaystyle \frac{1}{n}, since \displaystyle n = \frac{n}{1}. A number times its reciprocal is always 1:
Intuition
Think in scoops. Figure 1 shows 3 cups of flour cut into quarter cups. A \displaystyle \frac{3}{4}-cup scoop takes three of those quarters at a time, and the twelve quarters make four groups, so \displaystyle 3 \div \frac{3}{4} = 4.
The question that unlocks the rule is: how many scoops fill one cup? Figure 2 answers it for a \displaystyle \frac{2}{3}-cup scoop. One cup is three thirds; a scoop takes two of them, and the third left over is half a scoop. So one cup holds \displaystyle \frac{3}{2} scoops, and \displaystyle \frac{3}{2} is the reciprocal of \displaystyle \frac{2}{3}.
This is no accident. A scoop of \displaystyle \frac{c}{d} cup takes c of the d equal pieces that make a cup, so one cup holds d pieces \div c pieces per scoop, which is \displaystyle \frac{d}{c} scoops. Once you know how many scoops fill one cup, any amount of flour holds that many scoops for each cup of it. Dividing by \displaystyle \frac{c}{d} and multiplying by \displaystyle \frac{d}{c} are the same thing.
Results
Dividing by a fraction is multiplying by its reciprocal. For whole numbers a, b, c, d with b, c and d not 0:
Two cases worth knowing on their own. Over a common denominator (n and c not 0), the denominators drop out:
And one cup holds exactly the reciprocal of the scoop size:
Why it is true
Here are three separate arguments. Each one alone is enough; together they show the rule from three sides.
By common denominators. Rewrite both fractions over the denominator bd (Equivalent fractions):
Now both amounts are counted in the same pieces, each \displaystyle \frac{1}{bd} of a cup. Asking how many groups of cb pieces fit into ad pieces is a whole-number question, and its answer is \displaystyle ad \div cb = \frac{ad}{bc}. That is exactly \displaystyle \frac{a}{b} \times \frac{d}{c}.
By multiplying both numbers by the same thing. A quotient does not change when both numbers are multiplied by the same number (other than 0). With whole numbers, 12 \div 4 = 3 and 24 \div 8 = 3: twice the flour and a scoop twice as big make the same number of scoops. The reason is the definition: if q \times b = a, then q \times (kb) = k(q \times b) = ka, using the commutative and associative properties, so q is also ka \div kb. Now multiply both \displaystyle \frac{a}{b} and \displaystyle \frac{c}{d} by \displaystyle \frac{d}{c}:
The reciprocal is chosen because it turns the divisor into 1, and dividing by 1 changes nothing.
By undoing. Multiplying by \displaystyle \frac{c}{d} is undone by dividing by \displaystyle \frac{c}{d}; that is what division means. It is also undone by multiplying by \displaystyle \frac{d}{c}, because
using the associative property. So check the claimed answer the way the definition asks, by multiplying it by the divisor:
So \displaystyle \frac{a}{b} \times \frac{d}{c} is a number that gives \displaystyle \frac{a}{b} when multiplied by \displaystyle \frac{c}{d}. It is the only one: multiplying a larger number by \displaystyle \frac{c}{d} gives a larger product, so two different numbers cannot both give \displaystyle \frac{a}{b}. Dividing by a number and multiplying by its reciprocal undo exactly the same thing, so they are the same operation. \blacksquare
Worked examples
The lesson is taught in two parts. Examples 1 to 3 belong to the first: what dividing by a fraction means, counted on a bar and then over a common denominator. Examples 4 to 6 belong to the second: why the count always equals multiplying by the reciprocal.
Counting scoops in whole cups
Paloma has 3 cups of flour and a measuring cup that holds \displaystyle \frac{3}{4} of a cup. How many full scoops does the flour make?
SolutionThe question is how many \displaystyle \frac{3}{4}-cup amounts fit in 3 cups, which is \displaystyle 3 \div \frac{3}{4}. Cut each cup into quarters, as in Figure 1. Three cups make 3 \times 4 = 12 quarter cups. Each scoop takes 3 of those quarters, and 12 quarters split into groups of 3 make 4 groups:
Check. By the definition, 4 scoops of \displaystyle \frac{3}{4} cup should make 3 cups: \displaystyle 4 \times \frac{3}{4} = \frac{12}{4} = 3. They do.
A scoop that does not fit evenly
Sione has 2 cups of rice and the same \displaystyle \frac{3}{4}-cup measure. How many scoops is that?
SolutionAgain cut into quarters (Figure 3). Two cups are 8 quarter cups. Groups of 3 quarters give two full scoops, using 6 quarters, and leave 2 quarters over.
The leftover is \displaystyle \frac{1}{2} a cup of rice, but the question counts scoops, not cups. A full scoop needs 3 quarters and only 2 are left, so the leftover is \displaystyle \frac{2}{3} of a scoop:
Notice that the answer is 8 quarters \div 3 quarters, which is \displaystyle \frac{8}{3}, with no picture needed once both amounts are in quarters.
Check. \displaystyle \frac{8}{3} \times \frac{3}{4} = \frac{24}{12} = 2 cups.
Dividing over a common denominator
Greta has \displaystyle \frac{3}{4} of a cup of flour and a \displaystyle \frac{2}{3}-cup scoop. How many scoops is that?
SolutionQuarters and thirds are different pieces, so first count both amounts in the same piece. Twelfths work for both (common denominators):
Now the question is how many groups of 8 twelfths fit in 9 twelfths. Because both are counted in the same unit, this is the same as asking how many 8s fit in 9, just as 9 apples hold 9 \div 8 groups of 8 apples:
So Greta’s flour fills one scoop and \displaystyle \frac{1}{8} of another. The answer is a little more than 1, which fits: \displaystyle \frac{3}{4} of a cup is a little more than \displaystyle \frac{2}{3} of a cup.
Check. \displaystyle \frac{9}{8} \times \frac{2}{3} = \frac{18}{24} = \frac{3}{4}.
How many scoops fill one cup
Yusuf has 4 cups of flour and a \displaystyle \frac{2}{3}-cup scoop. Find the number of scoops by first finding how many scoops fill one cup.
SolutionFigure 2 shows one cup cut into thirds. The scoop takes two thirds; the third left over is half of what a scoop needs. So one cup holds
Every cup of flour holds the same \displaystyle \frac{3}{2} scoops, so 4 cups hold 4 times as many:
The division became a multiplication by \displaystyle \frac{3}{2}, the reciprocal of \displaystyle \frac{2}{3}, because the reciprocal is the number of scoops in one cup.
Check. By common denominators: \displaystyle 4 = \frac{12}{3}, and 12 thirds \div 2 thirds is 6. Both methods agree.
Making the divisor equal to one
Akosua has \displaystyle \frac{5}{6} of a cup of rice and a \displaystyle \frac{3}{4}-cup scoop. Find \displaystyle \frac{5}{6} \div \frac{3}{4} by multiplying both numbers by the same thing.
SolutionA quotient is unchanged when both numbers are multiplied by the same number other than 0 (twice the rice in a scoop twice as big gives the same count). Choose the number that makes the scoop exactly 1, which is \displaystyle \frac{4}{3}, since \displaystyle \frac{3}{4} \times \frac{4}{3} = 1:
Dividing by 1 changes nothing, so the answer is \displaystyle \frac{10}{9}, one scoop and \displaystyle \frac{1}{9} of another. The steps amount to multiplying \displaystyle \frac{5}{6} by \displaystyle \frac{4}{3}, the reciprocal of the divisor.
Check. By common denominators, \displaystyle \frac{10}{12} \div \frac{9}{12} = 10 \div 9 = \frac{10}{9}.
Proving an answer by undoing
Jonah has \displaystyle \frac{7}{8} of a cup of flour and a \displaystyle \frac{2}{5}-cup scoop. He claims the number of scoops is \displaystyle \frac{7}{8} \times \frac{5}{2}. Show that he is right using only the definition of division.
SolutionBy the definition, \displaystyle \frac{7}{8} \div \frac{2}{5} is the number that gives \displaystyle \frac{7}{8} when multiplied by \displaystyle \frac{2}{5}. So multiply Jonah’s number by \displaystyle \frac{2}{5} and see what comes out. Regroup with the associative property so the reciprocals meet:
Jonah’s number does the job, and only one number can, so
The flour makes two full scoops and \displaystyle \frac{3}{16} of a third. The same regrouping works for any divisor, which is the undoing argument of the proof.
Boundaries
The divisor cannot be 0. Zero has no reciprocal, since 0 times any number is 0, never 1. Likewise 5 \div 0 would need a number q with q \times 0 = 5, and there is none. Dividing 0 is fine: \displaystyle 0 \div \frac{3}{4} = 0, since no flour makes no scoops.
Only the divisor is flipped. Division is not commutative. \displaystyle \frac{3}{4} \div \frac{2}{3} = \frac{9}{8}, but \displaystyle \frac{2}{3} \div \frac{3}{4} = \frac{8}{9}: swapping the two numbers turns the answer into its reciprocal.
The answer counts scoops, not cups. In \displaystyle 2 \div \frac{3}{4} the leftover rice is \displaystyle \frac{1}{2} cup, yet the quotient’s fraction part is \displaystyle \frac{2}{3}, because it measures part of a scoop.
Dividing does not always make smaller. When the number being divided is greater than 0, dividing it by a number between 0 and 1 gives more than you started with (a small scoop is used many times); dividing by a number greater than 1 gives less; dividing by 1 changes nothing. Dividing 0 is the exception: 0 divided by any number other than 0 is 0, never more or less.
Mixed numbers must be rewritten first. The rule is stated for fractions \displaystyle \frac{a}{b}. A mixed number such as 2\tfrac{1}{2} is a sum, not a single fraction, so it has to become \displaystyle \frac{5}{2} before anything is flipped; that is the subject of Dividing mixed numbers.
Where this goes next
With the reason in hand, dividing by multiplying by the reciprocal turns the rule into a fast routine and practices it. After that, the same division finds a whole from a known part of it in finding the whole when only a fraction of it is known, and word problems that divide fractions asks which situations call for it.
The undoing argument reaches much further. Solving \displaystyle \frac{2}{5}x = \frac{7}{8} in algebra means multiplying both sides by \displaystyle \frac{5}{2}, which is the same step as Example 6 and is licensed by the properties of equality. Later, dividing by any number other than zero, negative ones included, is defined as multiplying by its reciprocal (Multiplying and dividing rational numbers).
Exercises
Exercises 1 to 10 go with the first part of the lesson (Examples 1 to 3), Exercises 11 to 17 with the second (Examples 4 to 6), and the reasoning group draws on both.
Part 1: counting scoops
Picture each amount on a bar, as in Figures 1 and 3, and count.
- How many \displaystyle \frac{1}{4}-cup scoops are in 2 cups of flour?
- How many \displaystyle \frac{2}{5}-cup scoops are in 2 cups of rice?
- How many \displaystyle \frac{3}{4}-cup scoops are in 1 cup of flour?
- Sione fills a jar with 3 cups of rice using a \displaystyle \frac{2}{3}-cup scoop. How many scoops?
- Find \displaystyle \frac{5}{6} \div \frac{1}{3} by cutting the bar into sixths.
Part 1: common denominators
Rewrite over a common denominator, then divide the numerators.
- \displaystyle \frac{7}{8} \div \frac{1}{4}
- \displaystyle \frac{2}{3} \div \frac{5}{6}
- \displaystyle \frac{9}{10} \div \frac{3}{5}
- \displaystyle \frac{5}{4} \div \frac{1}{3}
- Paloma has \displaystyle \frac{1}{2} cup of rice left and a \displaystyle \frac{3}{4}-cup scoop. How much of a scoop does the rice fill?
Part 2: the reciprocal
- Write the reciprocal of \displaystyle \frac{5}{7}, of 4 and of \displaystyle \frac{1}{9}.
- How many \displaystyle \frac{3}{5}-cup scoops fill one cup? Use that to find how many fill 6 cups.
- Find \displaystyle \frac{4}{9} \div \frac{2}{3} by multiplying both numbers by the number that makes the divisor 1.
- Find \displaystyle \frac{5}{8} \div \frac{5}{6} twice: over a common denominator, and by multiplying by the reciprocal. Do the answers agree? Use the common denominator 8 \times 6 to explain why they must.
- Yusuf says \displaystyle \frac{3}{10} \div \frac{3}{4} = \frac{2}{5}. Check his answer by multiplying, as in the definition.
- Akosua says \displaystyle \frac{4}{5} \div 2 = \frac{8}{5}. Check by multiplying, and give the correct quotient.
- Without working them out, say whether each quotient is larger than, smaller than or equal to \displaystyle \frac{5}{6}, then work them out: (a) \displaystyle \frac{5}{6} \div \frac{2}{3}, (b) \displaystyle \frac{5}{6} \div \frac{3}{2}, (c) \displaystyle \frac{5}{6} \div 1.
For more practice with the rule, see Dividing by multiplying by the reciprocal.
Reasoning
- Jonah says dividing always makes a number smaller. Use \displaystyle 2 \div \frac{1}{2} and a measuring cup to explain when he is wrong.
- Explain why \displaystyle 12 \div \frac{3}{4} must equal 48 \div 3, and find the value.
- Greta works out \displaystyle \frac{2}{3} \div \frac{3}{4} by flipping the first fraction and gets \displaystyle \frac{3}{2} \times \frac{3}{4} = \frac{9}{8}. Show by multiplying that \displaystyle \frac{9}{8} is not the quotient, and find the right one.
- Explain why 0 has no reciprocal, and what that says about dividing by 0.
- Show that \displaystyle \frac{a}{b} \div \frac{c}{d} and \displaystyle \frac{c}{d} \div \frac{a}{b} are always reciprocals of each other (all of a, b, c, d are whole numbers other than 0). What does that mean for two scoops of different sizes?
- Explain, by counting how many \displaystyle \frac{1}{5}-cup scoops fill one cup, why dividing any amount of flour by \displaystyle \frac{1}{5} gives the same answer as multiplying it by 5.
Answers
- 8. Two cups are 8 quarter cups, and each scoop is one of them.
- 5. Two cups are 10 fifths; groups of 2 fifths make 5 scoops.
- \displaystyle \frac{4}{3}. One cup is 4 quarters; one scoop uses 3, and the 1 quarter left is \displaystyle \frac{1}{3} of a scoop, so \displaystyle 1\tfrac{1}{3} = \frac{4}{3}.
- \displaystyle \frac{9}{2}, that is 4\tfrac{1}{2} scoops. Three cups are 9 thirds; four scoops use 8, and the 1 third left is half a scoop.
- \displaystyle \frac{5}{2}. \displaystyle \frac{1}{3} = \frac{2}{6}; 5 sixths make two groups of 2 sixths with 1 sixth, half a group, left over.
- \displaystyle \frac{7}{2}. \displaystyle \frac{7}{8} \div \frac{2}{8} = 7 \div 2 = \frac{7}{2}.
- \displaystyle \frac{4}{5}. \displaystyle \frac{4}{6} \div \frac{5}{6} = 4 \div 5 = \frac{4}{5}.
- \displaystyle \frac{3}{2}. \displaystyle \frac{9}{10} \div \frac{6}{10} = 9 \div 6 = \frac{3}{2}.
- \displaystyle \frac{15}{4}. \displaystyle \frac{15}{12} \div \frac{4}{12} = 15 \div 4 = \frac{15}{4}.
- \displaystyle \frac{2}{3} of a scoop. \displaystyle \frac{2}{4} \div \frac{3}{4} = 2 \div 3 = \frac{2}{3}.
- \displaystyle \frac{7}{5}, \displaystyle \frac{1}{4} and 9. Each times its number gives 1: for example \displaystyle \frac{5}{7} \times \frac{7}{5} = 1 and \displaystyle \frac{1}{9} \times 9 = 1.
- \displaystyle \frac{5}{3} scoops in one cup, so \displaystyle 6 \times \frac{5}{3} = 10 scoops in 6 cups. Check: \displaystyle 10 \times \frac{3}{5} = 6.
- \displaystyle \frac{2}{3}. Multiply both by \displaystyle \frac{3}{2}: \displaystyle \left(\frac{4}{9} \times \frac{3}{2}\right) \div 1 = \frac{12}{18} = \frac{2}{3}.
- Both give \displaystyle \frac{3}{4}: \displaystyle \frac{15}{24} \div \frac{20}{24} = \frac{15}{20} = \frac{3}{4}, and \displaystyle \frac{5}{8} \times \frac{6}{5} = \frac{30}{40} = \frac{3}{4}. They must agree: over the denominator 48, \displaystyle \frac{5}{8} = \frac{30}{48} and \displaystyle \frac{5}{6} = \frac{40}{48}, so the count is \displaystyle 30 \div 40 = \frac{30}{40}, the same numerator 5 \times 6 and denominator 8 \times 5 that multiplying by the reciprocal gives.
- He is right. \displaystyle \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}, the number being divided.
- \displaystyle \frac{2}{5}, not \displaystyle \frac{8}{5}. \displaystyle \frac{8}{5} \times 2 = \frac{16}{5}, not \displaystyle \frac{4}{5}. Dividing by 2 is multiplying by \displaystyle \frac{1}{2}: \displaystyle \frac{4}{5} \times \frac{1}{2} = \frac{2}{5}.
- (a) Larger, since \displaystyle \frac{2}{3} < 1: \displaystyle \frac{5}{6} \times \frac{3}{2} = \frac{5}{4}. (b) Smaller, since \displaystyle \frac{3}{2} > 1: \displaystyle \frac{5}{6} \times \frac{2}{3} = \frac{5}{9}. (c) Equal: \displaystyle \frac{5}{6}.
- \displaystyle 2 \div \frac{1}{2} = 4: a half-cup scoop fits into 2 cups four times. Dividing by a number less than 1 asks how many small pieces fit, and that is more than the amount itself. Dividing a number greater than 0 makes it smaller only when the divisor is greater than 1.
- 16. Multiplying both numbers by 4 does not change the quotient (four times the flour in a scoop four times as big), and it turns \displaystyle 12 \div \frac{3}{4} into 48 \div 3 = 16.
- \displaystyle \frac{9}{8} \times \frac{3}{4} = \frac{27}{32}, not \displaystyle \frac{2}{3}, so \displaystyle \frac{9}{8} fails the definition. The right quotient is \displaystyle \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}, and \displaystyle \frac{8}{9} \times \frac{3}{4} = \frac{24}{36} = \frac{2}{3}.
- A reciprocal of 0 would be a number that gives 1 when multiplied by 0, but 0 times anything is 0. In the same way, 5 \div 0 would need q \times 0 = 5, so no quotient exists: dividing by 0 is undefined.
- \displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} and \displaystyle \frac{c}{d} \div \frac{a}{b} = \frac{cb}{da}, and \displaystyle \frac{ad}{bc} \times \frac{bc}{ad} = 1. If a large scoop holds \displaystyle \frac{3}{2} small scoops, then a small scoop is \displaystyle \frac{2}{3} of a large one.
- A \displaystyle \frac{1}{5}-cup scoop fits into one cup 5 times, since a cup is 5 fifths. Every cup of flour holds the same 5 scoops, so the number of scoops is 5 times the number of cups: \displaystyle x \div \frac{1}{5} = x \times 5. Check by the definition: \displaystyle (x \times 5) \times \frac{1}{5} = x.