Ratios, Fractions & Scale Factor അംശബന്ധം · തോത്
Similar triangles are all about comparing lengths. "This side is 1½ times that side", "the small triangle is ⅔ of the big one", "the scale factor is 2.5" – these are the sentences you will read and write all through Chapter 6. Let us make sure they are crystal clear.
1. What is a ratio?
A ratio (അംശബന്ധം) compares two quantities by division. If a stick is 6 cm and another is 4 cm, the ratio of their lengths is 6 : 4, which is the fraction 64 = 32 = 1.5. Two things to remember:
- Both lengths must be in the same unit before you compare (2 m : 50 cm is 200 : 50 = 4 : 1).
- A ratio has no unit – it is just a number. 6 : 4, 3 : 2, 1.5 : 1 and 1.5 all say the same thing.
2. "1½ times" and "⅔ of" – two ways to say the same thing
If the big stick is 6 cm and the small one is 4 cm:
- The big one is 1½ times the small one: 4 × 1½ = 6.
- The small one is ⅔ of the big one: 6 × ⅔ = 4.
1½ and ⅔ are reciprocals: 1½ = 32 and ⅔ = 23; flip one to get the other, and their product is 1. Going small → big multiplies by k; going big → small multiplies by 1/k.
3. Scale factor തോത്
When every length of a figure is multiplied by the same number, that number is the scale factor. Enlarging a photo by scale factor 2 doubles its width and its height; the shape stays the same. If only the width is stretched, the picture is distorted – the shape changes, and the scale factor idea no longer applies.
e.g. k = 2: everything doubles
a congruent copy
e.g. k = ½: everything halves
4. Proportions and cross-multiplication
A proportion is a statement that two ratios are equal: ab = cd. Multiply both sides by b and by d and you get a × d = b × c – the "cross-multiplication" rule. It is the main tool for finding an unknown side in similar triangles.
5. Scales in real life
A map scale of 1 : 50 000 means 1 cm on the map = 50 000 cm = 500 m in the real world. So 3 cm on the map is 3 × 500 m = 1500 m = 1.5 km. The map is a "similar" picture of the land with scale factor 1/50 000.
A model car at scale 1 : 24 is a real car shrunk with scale factor 1/24. A 4.5 m long car becomes 450 ÷ 24 = 18.75 cm. Architects' building models, toy trains, model aircraft and the miniature Nehru-Trophy snake boats sold in Alappuzha all use a fixed scale factor so every part keeps the right proportions.
6. Percentage change is a scale factor too
"Increase by 25 %" means multiply by 1 + 0.25 = 1.25. "Reduce by 20 %" means multiply by 1 − 0.20 = 0.8. So percentage changes are just scale factors in disguise.
When two triangles are similar, every pair of matching sides has the same ratio – the scale factor k. Finding a missing side is always: write a proportion, cross-multiply, solve. Area, on the other hand, changes by k² (you will see why in Section 2).
7. Check yourself
- Ratio a : b = the fraction a/b; same units, no unit in the answer; simplify like a fraction.
- "k times" one way is "1/k of" the other way – reciprocals.
- Scale factor: every length × k. k > 1 enlarges, k < 1 reduces, k = 1 keeps the size.
- Proportion a/b = c/d ⇔ a·d = b·c (cross-multiply) – the tool for missing sides.
- Map scales, models and percentage changes are all scale factors.