HOW IT WORKS
Two boats race down the Nile. Boat A has finished 2/3 of the trip. Boat B has finished 3/5 of the trip. Which boat is farther along?
Step 1 — Notice the problem. Thirds and fifths are different-sized pieces. You can't just compare the tops.
Step 2 — Make the bottoms the same. Multiply the bottoms: 3 × 5 = 15.
Step 3 — Rewrite each one. 2/3 = 10/15 (× 5 top and bottom). 3/5 = 9/15 (× 3 top and bottom).
Step 4 — Compare the tops. 10/15 > 9/15, so Boat A (2/3) is farther along.
The trap: a bigger bottom number does not mean a bigger fraction. 1/8 is smaller than 1/3 — more pieces means each piece is smaller.
1.GUIDED
In a rowing race, Amir has rowed 3/4 of the way across the river and Sana has rowed 5/8 of the way. Who is farther across?
Same bottom number for both: 8 works, because 4 goes into 8.
Rewrite 3/4 into eighths: 3/4 = /8 (5/8 already has an 8)
Compare the tops: vs 5
Who is farther across?
Answer:
Did you know? The Nile flows north — it starts in the mountains and empties into the sea at the top of the map.
2.
A river lock has two gauges. One says the lock is 1/2 full. The other says 3/6 full. Are they showing the same amount of water, or different?
SHOW YOUR WORK
Answer:
Did you know? A lock is a water elevator — it fills and empties to lift boats between two river heights.
3.
Two streams feed a river. The first stream carries 2/5 of the river's water. The second carries 1/2 of it. Which stream carries more water?
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Answer:
Did you know? Small streams that join a bigger river are called tributaries — the Amazon has over 1,000 of them.
4.
Sana says 1/3 of the pond is frozen. Bilal says 1/4 is frozen. Whose fraction is the bigger amount? (Look at the pieces, not the bottom number.)
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Answer:
Did you know? Ice floats because water gets lighter when it freezes, so the ice stays on top.
HOW IT WORKS
A riverboat covers 1/2 of its journey on the first day and 1/3 more on the second day. How much of the journey is done?
Step 1 — Notice the problem. Halves and thirds are different-sized pieces. You cannot add 1 + 1, because the pieces aren't the same size.
Step 2 — Find a size they can both become. Multiply the bottoms: 2 × 3 = 6. Cut everything into sixths.
Step 3 — Rewrite each fraction. 1/2 = 3/6 (multiply top and bottom by 3). 1/3 = 2/6 (multiply top and bottom by 2).
Step 4 — Now add. 3/6 + 2/6 = 5/6. The bottom number does not change — you're just counting sixths.
The trap: 1/2 + 1/3 is not 2/5. Never add the bottoms. Multiplying the two bottoms together always gives a size they share, even if it isn't the smallest one.
1.GUIDED
Amir paddles 1/4 of a river in the morning and 2/5 of it in the afternoon. What fraction of the river has he paddled in all?
Common bottom number: 4 × 5 =
Rewrite: 1/4 = and 2/5 =
Add the tops, keep the bottom:
Answer:
Did you know? The Amazon carries more water than the next seven biggest rivers on Earth combined.
2.
A dam releases 3/8 of its water in the morning and 1/4 of its water in the evening. What fraction of the dam's water was released that day?
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Answer:
Did you know? The water held back by a big dam can be so heavy it slightly changes the speed the Earth spins.
3.
A boat travels 3/4 of a mile up one river, then 3/5 of a mile up another. How far did it go in total? Is that more or less than 1 mile?
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Answer:
Did you know? A mile was originally 1,000 paces of a Roman soldier — mille passus means "a thousand steps."
4.
A reservoir is 5/6 full. During a dry week, a town uses 1/4 of the reservoir. What fraction of the reservoir is left? (Same steps — but subtract this time.)
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Answer:
Did you know? A reservoir is just a man-made lake behind a dam, kept as a backup supply of drinking water.
3rd Grade (comparing fractions)
1. Amir is farther across.
Bottom 8 (4 goes into 8). 3/4 = 6/8. Compare tops: 6 vs 5 → 6/8 > 5/8, so Amir (3/4).
2. The same amount.
1/2 = 3/6 (multiply top and bottom by 3). Both gauges show the same water — 3/6 is just 1/2 written in sixths.
3. The second stream (1/2) carries more.
Bottom 10. 2/5 = 4/10 and 1/2 = 5/10. 5/10 > 4/10, so the 1/2 stream carries more.
4. Sana (1/3) is the bigger fraction.
Bottom 12. 1/3 = 4/12 and 1/4 = 3/12. 4/12 > 3/12, so 1/3 > 1/4. The bottom got bigger but the fraction got smaller — that's the trap.
4th Grade (unlike denominators)
1. 13/20 of the river.
Common bottom 4 × 5 = 20. 1/4 = 5/20, 2/5 = 8/20. 5/20 + 8/20 = 13/20.
2. 5/8 of the dam's water.
8 already works (4 divides into 8). 1/4 = 2/8. 3/8 + 2/8 = 5/8. Good one to point out you don't always have to multiply the bottoms.
3. 27/20 miles = 1 7/20 miles — more than 1 mile.
Common bottom 4 × 5 = 20. 3/4 = 15/20, 3/5 = 12/20. 15/20 + 12/20 = 27/20 = 1 whole and 7/20. Since 27 > 20, it's more than a mile.
4. 7/12 of the reservoir.
Common bottom 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 − 3/12 = 7/12. Same procedure as adding — only the last step changes.