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Rates Rates

Rates: how fast something changes

A rate tells you “how much per unit of time.”

rate = change ÷ time

Example 1: money rate (hourly pay)

If you earn $20 per hour, your pay rate is $20/hr.

That means your earnings go up by $20 every 1 hour.

money ($) hours (h) 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 120 140 160 (1, 20) (3, 60) E = 20h

Practice (with solutions)

1) If the rate is $20/hr, how much after 3 hours?

E = 20 × 3 = $60

2) How much after 4.5 hours?

E = 20 × 4.5 = $90

3) How many hours to earn $150?

150 ÷ 20 = 7.5 hours

Example 2: water tank rate (L/min)

A tank’s volume can change at a rate like +5 L/min (filling) or −3 L/min (draining).

Filling (positive rate) Draining (negative rate) V (L) V (L) time (min) time (min) 0 10 20 30 40 50 60 0 4 8 10 (0, 10) (8, 50) V = 10 + 5t 0 10 20 30 40 0 5 10 15 (0, 40) (6, 22) V = 40 − 3t

Practice (with solutions)

1) A tank starts at 10 L and fills at +5 L/min. How much after 8 minutes?

V = 10 + 5(8) = 50 L

2) A tank starts at 40 L and drains at −3 L/min. How much after 6 minutes?

V = 40 − 3(6) = 22 L

3) Same tank: when will it be empty?

0 = 40 − 3t → 3t = 40 → t = 40/3 ≈ 13.3 min

Quick check

If the rate is positive, the quantity is increasing. If the rate is negative, it’s decreasing.

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