Problem 1 · 2020 AMC 8
Easy
Ratios, Rates & Proportions
ratioproportion
Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as lemon juice. He uses 3 cups of lemon juice. How many cups of water does he need?
Show answer
Answer: E — 24 cups.
Show hints
Hint 1 of 2
Lemon juice is the “smallest” ingredient and everything is measured against it. So instead of two separate steps, ask: how many times bigger is water than lemon juice?
Still stuck? Show hint 2 →
Hint 2 of 2
When one thing scales another which scales a third, the scale factors multiply. Water is 4× sugar and sugar is 2× lemon, so water is 4 × 2 = 8 times the lemon juice.
Show solution
Approach: multiply the scale factors into one jump
- Chained scalings multiply: water is 4× sugar and sugar is 2× lemon, so water is 4 × 2 = 8 times the lemon juice — one jump instead of two.
- With 3 cups of lemon juice, water = 8 × 3 = 24 cups.
- You'll see this again as: any “A is k times B, B is m times C” chain collapses to “A is k·m times C.” Gear ratios and unit conversions work the same way.
Another way — one step at a time: lemon → sugar → water (MAA):
- Sugar is twice the lemon juice: 2 × 3 = 6 cups.
- Water is four times the sugar: 4 × 6 = 24 cups.
Mark:
· log in to save