Problem
Simplify complex fractions
A complex fraction has a fraction inside it. Four are given: 1 over 2/7; 8 over 4/3; 10/3 over 2/9; and 1 + 1/(1 + 1/2). The last has a fraction buried two layers down. Rewrite each as a single ordinary number.
Reasoning · Grade 6-1 Fraction and Decimal Calculations
Your answer
Watch out Layers tempt you to lump everything together, but a number outside must stay outside.
How to solve
Strategy Organize Information in More Ways — A stacked fraction looks like a brand-new object, but it is only an old one wearing a different costume: rewriting the bar as a division sign turns every one of these into a fraction-divided-by-fraction question that is already familiar. That rewrite is tool 15. Part (4) has two bars, so it needs tool 7 — deal with the innermost fraction first, get a single number underneath, and only then handle the outer bar. After doing (1) to (3) the same shape shows up every time, and tool 5 turns that into a shortcut worth remembering: the outer numbers end up on top and the inner numbers end up underneath.
1STEP 1
Read the long bar as a division sign
The long bar is a division sign.
□/△ = □ ÷ △
Once you know the bar is only a division sign in disguise, a complex fraction stops being a new kind of number and becomes a division you have done many times before.
5.NF.B.3Organize Information In More WaysThe long bar is a division sign, so a fraction stacked on a fraction is one number divided by another.
Why?
A fraction names how many equal shares of one whole you are taking, which is exactly what sharing out by division does.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
Why?
Dividing by a fraction is undone by multiplying by it, so the stacked bar can be turned into a multiplication.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
(1) Divide 1 by two-sevenths
(1) flips and multiplies to 3 1/2.
1/2/7 = 1 ÷ 2/7 = 1 × 7/2 = 7/2 = 3 1/2
Asking how many two-sevenths fit inside one whole is the same as asking how many pieces of size 2/7 make a whole, and three and a half of them do — so the answer being bigger than 1 is exactly right.
6.NS.A.1Organize Information In More Ways3STEP 3
(2) Divide 8 by four-thirds
(2) comes to 6.
8/4/3 = 8 ÷ 4/3 = 8 × 3/4 = 6
Cancelling before multiplying keeps the numbers small — 8 times 3 over 4 becomes 2 times 3 — and it is the same skill used for any fraction-of-a-whole-number question.
6.NS.A.1Organize Information In More Ways4STEP 4
(3) Divide ten-thirds by two-ninths
(3) comes to 15.
10/3/2/9 = 10/3 ÷ 2/9 = 10/3 × 9/2 = (5 × 3)/(1 × 1) = 15
Two-ninths is a very small piece and ten-thirds is more than three wholes, so it takes a lot of those pieces to fill it — a large whole-number answer like 15 is what you should expect.
6.NS.A.1Organize Information In More Ways5STEP 5
Notice the shortcut hiding in (1), (2) and (3)
A shortcut hides here: outers over inners.
a/b/c/d = (a × d)/(b × c), 10/3/2/9 = (10 × 9)/(3 × 2) = 90/6 = 15
Doing three problems the same way and then naming the pattern turns a three-step procedure into a one-glance rule, and the rule is safe because it is just flip-and-multiply written out.
6.NS.A.1Look For A Pattern6STEP 6
(4) Work the innermost layer first
(4) is worked from the innermost layer.
1 + 1/(1 + 1/2) = 1 + 1/3/2 = 1 + 2/3 = 1 2/3
A tower of fractions is just several small problems stacked up, and solving the deepest one first replaces it by a single number, so the tower gets one storey shorter each time.
5.OA.A.1Identify Subproblems7STEP 7
Check that the outside 1 was not swept into the fraction
Keeping the outside 1 gives 1 2/3.
Deciding which pieces belong above the main bar and which sit outside it is the whole difficulty in a stacked expression, and a quick size check catches the mistake immediately.
5.OA.A.1Identify SubproblemsAnswer
3 1/2, 6, 15, 1 2/3
10/3 × 9/2 = 15
Each answer is the right size. In (1) you are counting how many 2/7s fit in 1; since 2/7 is a bit less than a third, about three and a half fit — matching 3 1/2. In (2) 4/3 is a bit more than 1, so the answer should be a bit less than 8, and 6 is. In (3) 2/9 is small and 10/3 ≈ 3.3 is over three wholes, so a big answer is expected, and 15 × 2/9 = 30/9 = 10/3 multiplies back correctly. In (4) the fraction added on is 2/3, which is less than 1, so the total lies between 1 and 2. Multiplying each answer back by its denominator returns the numerator every time, which is the surest check of all.
Takeaway
The long fraction bar is just a division sign, so a fraction inside a fraction is only a division you already know — start with the deepest bar and work your way out.
- Read the long bar as a division sign
- (1) Divide 1 by two-sevenths
- (2) Divide 8 by four-thirds
- (3) Divide ten-thirds by two-ninths
- Notice the shortcut hiding in (1), (2) and (3)
- (4) Work the innermost layer first
- Check that the outside 1 was not swept into the fraction