3.2 Proportions & ratios: rate, speed-distance-time, scales, unit pricing, work-rate

Ratios, Proportions, and Rates

A ratio compares two quantities; a proportion sets two ratios equal. On the WorkKeys test these ideas power a huge share of questions: figuring pay per hour, speed, scale drawings, best buys, mixing solutions, and shared workloads. The unifying skill is the proportion - build one, then cross-multiply to solve.

Setting up a proportion

A proportion says a/b = c/d. Cross-multiplying gives a x d = b x c, and you solve for the unknown.

Example. A machine fills 150 bottles in 6 minutes. At the same rate, how many bottles in 20 minutes? Set up bottles over minutes on both sides:

150/6 = x/20.

Cross-multiply: 6x = 150 x 20 = 3,000, so x = 3,000 / 6 = 500 bottles.

Keep the same units in the same positions - bottles on top, minutes on bottom, on both sides. Mismatched setups are the top cause of wrong answers here.

Unit rates

A unit rate expresses how much per one - miles per gallon, dollars per pound, parts per hour. Divide the first quantity by the second.

Example. A worker assembles 96 units in 8 hours. The unit rate is 96 / 8 = 12 units per hour.

Example - pay rate. An employee earns $522 for a 36-hour week. The hourly rate is $522 / 36 = $14.50 per hour. Unit rates like this let you compare two jobs, two suppliers, or two speeds on equal footing.

Unit rates make comparison easy: once everything is stated per one, the numbers line up directly and the larger or smaller value is obvious.

Speed, distance, and time

Speed is just a unit rate for distance: speed = distance / time. The same relationship rearranges to distance = speed x time and time = distance / speed.

Example - speed. A delivery van covers 276 miles in 6 hours. Speed = 276 / 6 = 46 miles per hour.

Example - time. At that 46 mph pace, how long to drive the next 138 miles? time = distance / speed = 138 / 46 = 3 hours.

Example - distance. A courier rides for 2.5 hours at 18 mph. distance = speed x time = 18 x 2.5 = 45 miles.

Watch the units: if speed is in miles per hour, time must be in hours. A trip of 90 minutes is 1.5 hours, not 90.

Map and blueprint scales

Scales are ratios between drawing size and real size. Read the scale, then convert with a proportion.

Example - blueprint. A floor plan uses the scale 1/4 inch = 1 foot. A hallway is drawn 6.5 inches long. What is the real length? Each inch equals 4 feet (because 1 / (1/4) = 4), so 6.5 x 4 = 26 feet.

Example - map. A map scale is 1 inch = 25 miles. Two towns are 3.4 inches apart on the map. Real distance = 3.4 x 25 = 85 miles.

Unit pricing - finding the best buy

To compare package sizes, reduce each to price per unit, then pick the smaller number.

Example. A store offers dish soap two ways: a 32-ounce bottle for $5.12, or a 20-ounce bottle for $3.40. Compute price per ounce:

• 32 oz: $5.12 / 32 = $0.16 per ounce
• 20 oz: $3.40 / 20 = $0.17 per ounce

The 32-ounce bottle is the better buy at $0.16 per ounce.

Mixing ratios

Mixing problems give a ratio of ingredients; split the total into parts.

Example. A cleaning concentrate is mixed with water in a 1:5 ratio (one part concentrate to five parts water). To make 24 gallons of solution, how much concentrate is needed? Total parts = 1 + 5 = 6. Each part = 24 / 6 = 4 gallons. Concentrate = 1 part = 4 gallons; water = 5 parts = 20 gallons. Check: 4 + 20 = 24. Correct.

Work-rate problems

When two people or machines work together, add their rates (jobs per hour), not their times.

Example. One painter can paint a room in 6 hours; a second painter can paint the same room in 3 hours. Working together, how long?

• Painter A's rate = 1/6 room per hour
• Painter B's rate = 1/3 room per hour
• Combined = 1/6 + 2/6 = 3/6 = 1/2 room per hour

At half a room per hour, the whole room takes 1 / (1/2) = 2 hours.

Example - additive output. Machine A stamps 40 parts per hour and Machine B stamps 60 parts per hour. Together they make 100 per hour, so a 500-part order takes 500 / 100 = 5 hours.

Test-day strategy

Label every quantity, keep units consistent, and set the proportion up before touching the calculator. For best-buy and speed problems, a quick estimate - about 46, not 460 - flags decimal-point slips. When a problem mixes units, such as minutes with an mph speed, convert first and only then compute. Cross-multiplication solves almost every ratio item, so make it your default move.