GED Mathematical Reasoning
Ratios, Rates & Proportions
A ratio compares two amounts: 3 cups of water to 1 cup of rice is 3 to 1. A rate is a ratio where the two things have different units, like 60 miles per 1 hour. A proportion is just two ratios set equal, which is how you scale a recipe up or a map distance down.
The words first
- ratio
- a comparison of two amounts, written 3:1 or 3 to 1 or 3/1
- rate
- a ratio comparing different units — miles per hour, dollars per pound
- unit rate
- the rate for exactly one — cost for ONE pound, miles in ONE hour
- proportion
- two equal ratios, like 3/1 = 12/4
The method
SETTING UP A PROPORTION — keep the units in the same spots
miles miles
—————— = ——————
hours hours
Miles on top on BOTH sides. Hours on the bottom on BOTH sides.
If you flip one side, the answer comes out wrong.
SOLVING IT — cross multiply
a c
— = — becomes a × d = b × c
b d
FINDING A UNIT RATE — divide by the number of units
$12 for 4 pounds → 12 ÷ 4 = $3 per pound
Worked all the way through
The problem
A car travels 150 miles on 6 gallons of gas. How far can it go on 10 gallons?
- Step 1 — Decide what goes on top.Miles on top, gallons on the bottom. Both sides must match.
- Step 2 — Write what you know on the left.150 miles / 6 gallons
- Step 3 — Write what you want on the right, using x for the unknown.150/6 = x/10
- Step 4 — Cross multiply.150 × 10 = 6 × x → 1500 = 6x
- Step 5 — Undo the multiplication by dividing both sides by 6.1500 ÷ 6 = 250, so x = 250
On the board
150/6 = x/10
miles on top, gallons on the bottom
150 × 10 = 6 × x
cross multiply
1500 = 6x
1500 ÷ 6 = 6x ÷ 6
divide BOTH sides by 6
x = 250
Answer
250 miles
A second one, in a different form
If 3 pounds of apples cost $5.25, what is the cost per pound?
5.25 ÷ 3
total ÷ how many = unit rate
1.75
$1.75 per pound
Answer
$1.75 per pound
The usual mistake
Reading a 3:5 ratio as 3 out of 5 rather than 3 out of 8.
Check yourself
Check the direction: more gas should mean more miles. 10 gallons is more than 6, and 250 is more than 150. Good.
Practice it
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