Two teams grow in the same design. When do they become EQUAL? And how does dividing find the answer? Click Next.
The Star Box designs: stars ★ in the middle (n² of them — they grow FAST). circles ○ = 4n (they grow steady, +4 each design).
| Design n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Stars ★ (n²) | 1 | 4 | 9 | 16 | 25 |
| Circles ○ (4n) | 4 | 8 | 12 | 16 | 20 |
Look down each column. Who has more — stars or circles?
The expressions come FROM the picture. Look at Design 3.
Test both expressions on Design 3:
stars: 3 × 3 = 9 ✓ circles: 4 × 3 = 12 ✓
Both match the picture. Now we trust the expressions: n² and 4n.
Gold bar = stars. Blue bar = circles. Taller bar is winning.
Circles start ahead. Stars grow faster. At n = 4 they are EQUAL — then stars take over. Every "when are they equal?" question is asking for this crossing point.
Drawing bars for every design is slow. Division compares two numbers in ONE step.
Try it with real numbers.
Design 2: 4 ÷ 8 = 0.5 → stars are only HALF of circles. Circles winning.
Design 4: 16 ÷ 16 = 1 → a TIE!
The rule of the compare machine:
answer less than 1 → the BOTTOM team is winning
answer = 1 → THE TIE — the two teams are EQUAL
answer more than 1 → the TOP team is winning
16 ÷ 16 = 1. Equal numbers ALWAYS divide to 1. That is the whole secret.
We built them from the picture: stars = n², circles = 4n. One division with n inside checks EVERY design at once.
Divide stars by circles:
n² ÷ 4n = n × n ÷ 4 × n = n ÷ 4
One n on top and one n on the bottom cancel each other — cross them out. A big scary division becomes tiny: n ÷ 4.
Ask for the tie. The tie means the division = 1:
n ÷ 4 = 1 → n = 4
Check: Design 4 → stars 4² = 16, circles 4×4 = 16. EQUAL ✓ The division found the crossing point — no drawing, no big table!
Example 2 — which design has stars 4 TIMES the circles?
Same expressions, same division: n² ÷ 4n = n ÷ 4
"4 times the circles" means the division = 4: n ÷ 4 = 4 → n = 16
Check Design 16 — and compare with the tie.
stars: 16² = 256 circles: 4×16 = 64 256 ÷ 64 = 4 ✓
At n = 4 the division was 1 → EQUAL.
At n = 16 the division is 4 → stars are 4 times the circles.
Only the LAST step changed: the 1 became a 4. Same recipe, new number.
Five small steps. This works on every "when are they equal?" problem.
1. Build the expression for each team from the picture (stars = n², circles = 4n).
2. Divide them: first team on top → n² ÷ 4n.
3. Cancel the n → n ÷ 4.
4. "EQUAL" means the division = 1 (and "4 times as many" means = 4).
5. Solve the tiny equation: n ÷ 4 = 1 → n = 4. That n is the design number!
The tie is 1, not 0. Equal teams divide to 1 (16 ÷ 16 = 1). Students write "= 0" and get lost.
Order matters. stars ÷ circles is not circles ÷ stars. Put the team named FIRST in the question on TOP.
Divide = compare. Answer 1 = tie. Cancel the n, then solve the tiny leftover.
A design has squares = n² and triangles = 5n. In which design are they EQUAL?
Answer: (B) Design 5.
1. Divide: n² ÷ 5n = n ÷ 5 (one n cancels).
2. The tie is 1: n ÷ 5 = 1 → n = 5.
3. Check: squares 5² = 25, triangles 5×5 = 25. EQUAL ✓