Day 3 — Ratios
Time: ~20 min · Topic: ratios
The problem
A lemonade recipe calls for 2 cups of lemon juice for every 8 cups of water. If you use 3 cups of lemon juice, how many cups of water do you need?
Hint: Try figuring out how many cups of water go with just 1 cup of lemon juice first.
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Find the ratio of lemon juice to water. For every 2 cups of juice, you need 8 cups of water. That simplifies to 1 cup of juice for every 4 cups of water (divide both by 2).
- Now multiply. If 1 cup of juice needs 4 cups of water, then 3 cups of juice needs 3 x 4 = 12 cups of water.
- Check your answer — 2:8 and 3:12 both simplify to 1:4. The ratio holds!
Answer: 12 cups of water
Why this matters
Ratios are the secret behind every perfect recipe — professional bakers use a ratio of 5:3:2 (flour:liquid:fat) to make almost any bread!
This day in math: The Mathematician Who Broke Infinity Into Pieces
April 23, 1858 · Max Planck
On April 23, 1858, Max Planck was born in Kiel, Germany. In 1900, he stunned the scientific world by proving that energy isn't continuous — it comes in tiny, discrete packets called "quanta." His deceptively simple equation E = hf, where energy equals Planck's constant times frequency, shattered centuries of assumption that nature behaves smoothly.
Why it matters: Planck's insight — that the universe is fundamentally discrete, not continuous — launched quantum mechanics and reshaped both physics and mathematics. Every piece of modern technology, from smartphones to MRI machines, traces back to this one equation.
Did you know?
There are more numbers between 0 and 1 than there are counting numbers (1, 2, 3, 4, ...) stretching to infinity. Not all infinities are the same size — some infinities are literally BIGGER than others!
In 1891, Georg Cantor proved this with his famous "diagonal argument." He showed that no matter how you try to pair up every decimal between 0 and 1 with a counting number, you can always build a new decimal that isn't on your list. This means the infinity of real numbers is "uncountable" — a fundamentally larger type of infinity than the "countable" infinity of 1, 2, 3, 4, and so on.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact