Day 1 — Linear Equations
Time: ~30 min · Topic: linear equations
The problem
Phone Plan A charges a $15 monthly fee plus $0.10 per text message. Plan B charges a $5 monthly fee plus $0.30 per text message. How many text messages would make both plans cost exactly the same?
Hint: Try writing a cost equation for each plan using the same variable, then set them equal to each other.
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
- Write equations for each plan. Let x = number of texts. Plan A: Cost = 15 + 0.10x. Plan B: Cost = 5 + 0.30x.
- Set the two costs equal: 15 + 0.10x = 5 + 0.30x
- Subtract 0.10x from both sides: 15 = 5 + 0.20x. Then subtract 5: 10 = 0.20x
- Divide both sides by 0.20: x = 50. Both plans cost $20.00 at 50 texts.
Answer: 50 text messages (at which point both plans cost $20.00)
Why this matters
Businesses use this exact same math — called "break-even analysis" — to figure out when a new product will start making profit!
This day in math: The Man Who Mapped Magnetic Fields
April 21, 1774 · Jean-Baptiste Biot
On April 21, 1774, French mathematician and physicist Jean-Baptiste Biot was born in Paris. In 1820, Biot and Félix Savart discovered the mathematical law governing how electric currents create magnetic fields — now known as the Biot-Savart Law. He also pioneered the study of light polarization through organic substances, establishing Biot's Law of Rotary Polarization.
Why it matters: The Biot-Savart Law is foundational to electromagnetism and is used today in everything from MRI machines to electric motor design. His work on light polarization laid the groundwork for modern optics and even helps chemists identify molecular structures.
Did you know?
Cicadas have evolved to emerge on prime number cycles — every 13 or 17 years — and mathematicians believe this is no coincidence. By syncing their life cycles to prime numbers, these insects gain a powerful survival advantage that's built on pure number theory.
Because prime numbers have no smaller divisors other than 1, a predator with a repeating 2-, 3-, 4-, or 5-year cycle will almost never sync up with a 13- or 17-year emergence. For example, a predator on a 4-year cycle would only overlap with a 17-year cicada once every 68 years (4 × 17). If cicadas emerged every 12 years instead, predators on 2-, 3-, 4-, or 6-year cycles would ALL line up repeatedly. Primes minimize these dangerous overlaps — evolution discovered number theory millions of years before we did.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact