Rates
A rate compares two quantities with different units, such as speed (km/h), price (TSh per kg) or a pay rate (TSh per hour). A rate is written "per" one unit of the second quantity.
A machine fills 360 bottles in 8 minutes. How many bottles in 30 minutes?
- Rate per minute.
- Multiply by 30 minutes.
Answer: 1 350 bottles.
Direct variation
Two quantities vary directly if one increases in the same ratio as the other: y = kx, where k is a constant. The graph is a straight line through the origin. Example: the cost of x kg of rice.
Inverse variation
Two quantities vary inversely if one increases as the other decreases so that their product is constant: y = k/x. The graph is a curve (a hyperbola). Example: the time to do a job and the number of workers.
| Type | Equation | Finding k | Graph |
|---|---|---|---|
| Direct | y = kx | k = y ÷ x | Straight line through the origin |
| Inverse | y = k/x | k = xy | Hyperbola |
| Joint | z = kxy | k = z ÷ xy | |
| Partial | y = a + bx | Two constants | Line not through the origin |
6 workers build a wall in 10 days. How long would 15 workers take?
- Days vary inversely with workers: workers × days is constant.
- Divide by 15.
Answer: 4 days.
Choose k and x. See how y changes in each kind of variation.
Key terms
- Rate
- A comparison of two quantities with different units.
- Direct variation
- y = kx.
- Inverse variation
- y = k/x.
- Constant of variation
- The fixed number k.
Summary
- A rate compares quantities with different units.
- Direct variation: y = kx, a line through the origin.
- Inverse variation: y = k/x, so xy is constant.
- Find k from one pair of values, then use it.