Before you begin
One sentence governs this entire quiz: pieces can only be joined when they are the same size. That is all addition of fractions is, and all subtraction too. Everything difficult here is just the work of making pieces match before combining them. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.
Same size pieces: add the count, keep the name
Three eighths and two eighths make five eighths. The eighths never change, because the denominator is a name, not a count — it tells what kind of piece you are holding. You would say three apples plus two apples is five apples, not five appleapples. The bottom number stays put.
The error worth naming out loud
The great temptation is to add straight across, top to top and bottom to bottom, so that 12 + 12 becomes 24. Follow that where it leads: two halves of a pizza would come to half a pizza. You would have eaten both halves and still be holding half.
Different sizes: re-cut until they match
To add 12 and 13, find a piece size that both can be rebuilt from. Sixths work: a half is three sixths, a third is two sixths. Now they match, so the counts can be added: 3 + 2 = 5, giving five sixths.
Any common denominator (a bottom number both fractions can be re-cut into) works; the least one keeps the numbers small, and Quiz 7 shows how to find it by listing multiples. When the bottoms share no factors, their product always works: for fifths and fourths, use twentieths.
Subtraction is the same rule
Match the pieces, then take away the count: 34 − 16 becomes 912 − 212 = 712. Nothing new is required.
Finishing the answer
Two habits complete every problem. Simplify, dividing top and bottom by any factor they share (Quiz 8's method), so the answer wears its plainest name. And if the top has grown larger than the bottom, convert to a mixed number, a whole number with a fraction beside it: 1312 means 13 ÷ 12, which is 1 1/12.
Mixed numbers: carrying and borrowing again
Add the whole parts and the fraction parts separately, then tidy up. For 2⅓ + 1½: the wholes make 3, and the fractions make 26 + 36 = 56, so the answer is 3 5/6. When the fraction part reaches a whole or more, carry it over to the whole number, exactly as Quiz 3 carried a ten.
Subtraction can require the other old trade. In 5¼ − 2¾ the fourths come up short, since you cannot take 3 fourths from 1 fourth. So break a whole into fourths, precisely as Quiz 3 broke a ten into ones:
So 5¼ becomes 4 and 54. Then 5 fourths less 3 fourths is 2 fourths, and 4 less 2 is 2: the answer is 2 2/4, which simplifies to 2½.
The guard, still on duty
Estimate before you trust. Round each fraction to 0, ½, or 1: for 56 + 14, that is about 1 + ¼, so the answer should sit a little above one. An answer of 610 — which is what adding across would give — is under one, and the guard catches it before the ink dries.
Three to study before you start
56 + 14. Guard: about 1 + ¼. Sixths and fourths both rebuild from twelfths: 1012 + 312 = 1312. The top outgrew the bottom, so convert: 13 ÷ 12 = 1 R 1, giving 1 1/12. Just above one, as the guard predicted.
134 + 223. Wholes: 1 + 2 = 3. Fractions in twelfths: 912 + 812 = 1712, which is 1 whole and 512. Carry that whole up: 3 + 1 = 4, so the answer is 4 5/12.
514 − 234. One fourth cannot give three, so break a whole: 5¼ becomes 4 and 54. Now 5 fourths less 3 fourths is 2 fourths, and 4 less 2 is 2: 2 2/4, simplified to 2½. Check by adding back: 2½ + 2¾ = 5¼, home again.
Now you
Work without a calculator, on paper. Give every answer in simplest form, and as a mixed number when the top outgrows the bottom. Questions 3 and 7 are multiple choice: choose the one best answer.
- Add: 38 + 28
- Subtract: 79 − 49
- Rosa worked 12 + 13 and wrote 25. What went wrong?
- A) Nothing; 2/5 is correct.
- B) She added straight across, but the bottom names the size of the piece and is never added; the halves and thirds must first be re-cut to match.
- C) She should have multiplied the bottoms and left the tops alone.
- D) She forgot to simplify her answer.
- Add: 12 + 13
- Subtract: 34 − 16
- Add: 56 + 14
▶ VideoA video rest stop, for anyone who has kept going this far.
Take two minutes to watch a seahorse father give birth, to some two thousand at once.
- Add: 213 + 112
- A) 3 2/5
- B) 3 5/6
- C) 4 1/6
- D) 3 1/6
- Subtract: 514 − 234
- A recipe calls for 34 cup of flour for the dough and 23 cup for the topping. How much flour in all?
- Brian has an 8-foot length of conduit. He cuts a piece 2 3/8 feet long and another 3 1/2 feet long. How much conduit is left?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 2 | Same denominator | Add or subtract the counts; the name of the piece never changes. |
| 3 | Naming the great error | Run the pizza test on any rule you are unsure of. |
| 4, 5, 6, 9 | Re-cutting to match | Find a size both can rebuild from, convert, then combine; simplify at the end. |
| 7 | Mixed numbers, carrying | Wholes with wholes, fractions with fractions, and carry only a whole you actually have. |
| 8, 10 | Mixed numbers, borrowing | Break one whole into the fraction's own pieces, exactly as Quiz 3 broke a ten. |
Eight or more right: the joining rule is yours — on to Quiz 11. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Cut two identical strips of paper with your teacher, one into halves and one into thirds, and try to combine them until the need for sixths becomes obvious. Then come back to these same ten.
Next in the Seahorse Series: Quiz 11 — Multiplying Fractions. A surprise waits there: multiplying needs no common denominator at all, and multiplying by a fraction makes things smaller. Both facts have good reasons, and the rectangle explains them.
Fellow travelers
All three of today's travelers come from Wikimedia Commons, where the pictures are shared rather than sold. A fitting source for a page about combining what people bring.
