Same look, same factor
Shrink a picture the right way and it still looks like itself.
A tiger picture
A picture of a tiger is 60 mm wide and 40 mm tall. A smaller copy is 30 mm × 20 mm, and it still looks like the same tiger. Another copy has 20 mm taken off both the width and the height.
Take 20 mm off the width and 20 mm off the height of the 60 × 40 picture. How will it look?
What this lesson covers
The idea
Changes to two quantities are proportional when both are multiplied by the same factor; changing both by the same amount through subtraction or addition does not in general keep them proportional.
A tiger picture
A picture of a tiger is 60 mm wide and 40 mm tall. A smaller copy is 30 mm × 20 mm, and it still looks like the same tiger. Another copy has 20 mm taken off both the width and the height.
Take 20 mm off the width and 20 mm off the height of the 60 × 40 picture. How will it look?
- Just like the first, only smaller
- Squashed or stretched
- Cannot tell
Make copies of picture A
The blue picture is A. The orange one is your copy. Make copies by multiplying both sides, or by taking off the same amount. Watch the factor for each side.
Same factor, same look
Changes to two quantities are proportional when both are multiplied by the same factor; changing both by the same amount through subtraction or addition does not in general keep them proportional.
Picture C: width 60 → 30 and height 40 → 20. Both are multiplied by 1/2, so C looks like A.
Picture B: 20 mm off both. Width 60 → 40 is × 2/3, but height 40 → 20 is × 1/2. Different factors, so B looks stretched.
Notes
Changes to two quantities are proportional when both are multiplied by the same factor; changing both by the same amount through subtraction or addition does not in general keep them proportional.
Check yourself
Picture A is 60 mm wide and 40 mm tall. Which copy will look just like A?
Picture D is 90 mm wide. What height (in mm) keeps the look of A (60 × 40)?
Answer: 60 mm
60 → 90 is × 1 1/2, so the height is 40 × 1 1/2 = 60 mm.
Rani prints A with 10 mm taken off both the width and the height. It is 50 × 30. Does it look like A?
Picture E is 60 mm × 60 mm. Picture A is 60 × 40. Why does E look different from A?
- 40 mm × 20 mm. Width × 2/3 but height × 1/2. The two sides changed by different factors.
- 30 mm × 20 mm — correct. Yes! Both sides are × 1/2: 60 → 30 and 40 → 20.
- 80 mm × 60 mm. Width × 4/3 but height × 3/2. Both gained 20 mm, but the factors differ.
- 50 mm × 30 mm. Width × 5/6 but height × 3/4. Both lost 10 mm, but the factors differ.
- Yes, the same amount was taken off both sides. Taking off the same amount does not keep the look. Check the factors.
- Yes, because both sides got smaller. Both getting smaller is not enough. They must change by the same factor.
- No: width × 5/6 but height × 3/4 — correct. Yes! 50/60 = 5/6 and 30/40 = 3/4. Different factors, so it looks changed.
- The width did not change, but the height became 1 1/2 times — correct. Yes! Width × 1 and height × 1 1/2: different factors.
- Both sides changed by 20 mm. The width did not change at all. Only the height grew, by 20 mm.
- It is a square, and squares never look like rectangles. The shape is not the reason. A 30 × 20 copy is a rectangle and looks right. What matters is the factors.