Problem 2 · AMC 8 Stretch
Core
Fractions, Decimals & Percents
Arithmetic & Operations
logical-reasoningvisual-representation
Reading a fraction as a COUNT of equal pieces, \(\frac{2}{3}\) means '2 thirds.' Using that idea (the bottom is the unit, the top is how many), what is \(\frac{2}{3}+\frac{5}{3}\)? Give your answer as a fraction.
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Answer: 7/3
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Hint 1 of 4
Think of '2 thirds' and '5 thirds' like '2 meters' and '5 meters.' When the unit (the bottom number) is the same, you just add how many you have.
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Hint 2 of 4
So add the tops and keep the bottom: \(2 + 5\) thirds.
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Hint 3 of 4
The fake rule 'add tops, add bottoms' would give \(\frac{1}{2}+\frac{1}{2}=\frac{2}{4}=\frac{1}{2}\), but two halves make a WHOLE — so that rule is wrong.
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Approach: Read the denominator as a fixed unit; add only the counts
- Why you never add the bottoms: the fake rule turns \(\frac{1}{2}+\frac{1}{2}\) into \(\frac{2}{4}=\frac{1}{2}\), but two halves make a whole \(= 1\). So 'add the bottoms' is false.
- Read the bottom as the NAME of the piece (the unit) and the top as HOW MANY. So \(\frac{2}{3}\) is '2 thirds' and \(\frac{5}{3}\) is '5 thirds.'
- With the same unit, adding is like \(2\text{ m} + 5\text{ m} = 7\text{ m}\): \(2\text{ thirds} + 5\text{ thirds} = 7\text{ thirds}\).
- So \(\frac{2}{3}+\frac{5}{3}=\frac{7}{3}\): add the tops, keep the bottom. Adding bottoms would secretly change the slice size mid-count.
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