Ratio is a way of comparing two quantities of the same kind by division. The ratio of a to b is written as a:b or a/b. Both quantities must have the same unit.
Simplifying Ratios
Divide both terms by their HCF to get the ratio in its simplest form.
Example: 24:36 = 2:3 (dividing both by HCF = 12).
Equivalent Ratios
Multiply or divide both terms by the same non-zero number.
2:3 = 4:6 = 6:9 = 10:15 are all equivalent ratios.
Proportion
Four quantities a, b, c, d are in proportion if a:b = c:d (or a/b = c/d).
Written as a:b::c:d, read as "a is to b as c is to d."
In a proportion: product of extremes = product of means.
So a:b::c:d means a x d = b x c.
Unitary Method
Find the value of one unit, then multiply to find the value of any number of units.
Find the ratio of 45 minutes to 2 hours.
Convert to same unit: 2 hours = 120 minutes.
Ratio = 45:120 = 3:8.
If the ratio of boys to girls in a class is 3:5 and there are 24 boys, how many girls are there?
3 parts = 24 boys, so 1 part = 8. Girls = 5 x 8 = 40.
Are 4, 12, 6, 18 in proportion?
Check: 4/12 = 1/3 and 6/18 = 1/3. Since they are equal, yes, 4:12::6:18.
Find the fourth proportional to 5, 10, and 15.
5:10::15:x. 5x = 10x15 = 150. x = 30.
If 8 books cost Rs 240, what is the cost of 13 books?
Cost of 1 book = 240/8 = Rs 30. Cost of 13 books = 13 x 30 = Rs 390.
Divide Rs 1,200 in the ratio 3:5.
Total parts = 8. First share = (3/8) x 1200 = Rs 450. Second share = (5/8) x 1200 = Rs 750.
A car travels 300 km in 5 hours. How far will it travel in 8 hours at the same speed?
Distance per hour = 300/5 = 60 km. In 8 hours: 8 x 60 = 480 km.
- Key Rule
- In proportion a:b::c:d:
- a and d are called extremes.
- b and c are called means.
- Extremes product = Means product: a x d = b x c.
Common mistakes
- Ratio requires the same unit: you cannot directly compare 50 paise and Rs 3 — convert first (300 paise).
- Ratio 3:5 does NOT mean 3 and 5 are the quantities; the quantities could be 6 and 10 or 9 and 15.
- Do not confuse ratio (comparison) with proportion (equality of two ratios).
Summary
Ratio compares two like quantities. Simplest form uses HCF. Proportion states two ratios are equal; cross-multiplication checks it. The unitary method solves practical ratio problems by first finding the unit value.