Comparing quantities with ratios and scaling with proportion
A ratio compares two quantities of the same kind by division and is written a : b. Ratios are simplified like fractions, so 18 : 24 becomes 3 : 4. To divide ₹560 between two people in the ratio 3 : 4, add the parts to get 7, find one part as ₹80, and give ₹240 and ₹320. Ratios can only be compared when the units match, so 50 cm : 2 m must be written as 50 : 200 before simplifying it to 1 : 4.
Four numbers are in proportion when a : b = c : d, which means the product of the extremes equals the product of the means, ad = bc. In direct proportion both quantities rise or fall together, so the cost of pens doubles when you buy twice as many. In inverse proportion one rises as the other falls: if 6 workers finish a job in 10 days, 12 workers need 5 days, because the product of workers and days stays at 60. The unitary method finds the value of one unit first and then scales up to the quantity required.
Why it matters
Ratios are used in recipes, map scales, mixing cement or medicines, sharing profits among partners and comparing prices per kilogram. Proportional reasoning is the base for percentages, similar triangles, speed and work problems, and for chemistry calculations with moles. Aptitude tests in many competitive exams include several ratio and mixture questions.
Key terms
- Ratio
- A comparison of two quantities of the same kind, showing how many times one contains the other.
- Direct proportion
- A relationship in which two quantities increase or decrease by the same factor.
- Inverse proportion
- A relationship in which one quantity increases as the other decreases, keeping their product fixed.
- Unitary method
- Finding the value of a single unit first and then multiplying to get the required amount.