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Prealgebra / LEVEL 7 · DIFFICULTY 3/5

Ratios, Units & Rates

Compare related quantities and coordinate changes across units.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST 7.1Ratios & Multiway SharesRead the lesson
  2. MINI QUEST 7.2Proportions & Unit ConversionsRead the lesson
  3. MINI QUEST 7.3Speed, Work & Combined RatesRead the lesson
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STAGE 7.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Ratios & Multiway Shares

Useful preparation: Equivalent Fractions & Comparison

Goal: Distinguish part-to-part ratios from part-to-whole fractions.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

State the order of the quantities

A ratio compares quantities in a stated order. Reversing the order reverses the ratio. Equivalent ratios scale all parts by the same factor.

red : blue = 3 : 5

The total counts all parts

A ratio 3:5 has eight total parts. The first category is 3/8 of the whole, not 3/5. More than two categories follow the same idea.

a : b : c has a + b + c total parts

Equalize a shared quantity

To combine two ratios that share a category, scale them until that category has the same number of parts in both.

A:B = 2:3, B:C = 6:5 ⇒ A:B:C = 4:6:5
WORKED EXAMPLE 1

Paints A, B and C are mixed in ratio 2:3:4. A batch is 72 ml. How much B is used?

  1. There are 9 total parts.
  2. Each part is 72/9 = 8 ml.
  3. B uses three parts.

24 ml

WORKED EXAMPLE 2

Cats:dogs = 3:4 and dogs:rabbits = 2:5. Find cats:rabbits.

  1. Scale the second ratio to dogs:rabbits = 4:10.
  2. Now the shared dog count matches.

3:10

Common pitfalls

Possible mix-up: For a 2:7 ratio, the first share is 2/7 of the total.

It is 2/(2 + 7) = 2/9 of the total.

Possible mix-up: Add the same number to all ratio parts to scale a mixture.

Equivalent ratios multiply all parts by the same factor.

Explain it to yourself

How can two mixtures have different amounts but the same composition?

Preview the eight practice prompts
  1. Red:blue beads = 3:5. There are 40 beads total. How many are red?
  2. Juice:water = 2:7. What fraction of the mixture is juice?
  3. A:B:C = 4:1:3 and B is 6. What is C?
  4. The ratio small:large boxes is 5:8. If there are 24 large boxes, how many small boxes are there?
  5. A:B = 2:3 and B:C = 9:4. If A = 18, find C.
  6. Two positive quantities have ratio 7:3 and differ by 28. What is the larger quantity?
  7. A garden assigns beds to herbs, greens and flowers in ratio 2:5:3. If flowers occupy 18 beds, how many beds are used in total? New context
  8. A trail mix has nuts:fruit = 3:2. It contains 18 cups of nuts. How many cups of fruit must be added, without changing nuts, to make nuts:fruit = 1:1? New context
Open stage 7.1 in the student workspace →

STAGE 7.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Proportions & Unit Conversions

Useful preparation: Ratios & Multiway Shares · Decimal Operations & Rounding

Goal: Use a constant multiplier and cancel units to connect measurements.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A proportion preserves a ratio

When y is directly proportional to x, y/x is constant. A fixed starting fee breaks direct proportionality unless it is accounted for separately.

y = kx

Carry units through the calculation

A conversion factor equals one expressed in different units. Arrange it so unwanted units cancel.

3 m × 100 cm/m = 300 cm

Convert powers of units too

Area and volume units involve squared or cubed lengths. Apply the length conversion to every dimension.

1 m² = 100² cm² = 10000 cm²
WORKED EXAMPLE 1

Five meters of cable cost $18 at a constant price per meter. What do 12 meters cost?

  1. The rate is 18/5 dollars per meter.
  2. Multiply by 12 meters.

43.20 dollars

WORKED EXAMPLE 2

Convert 0.7 square meters to square centimeters.

  1. One meter is 100 centimeters.
  2. Square the conversion factor: 10000.

7000 cm²

Common pitfalls

Possible mix-up: A fixed-fee price is directly proportional to quantity.

A fixed fee makes the ratio vary.

Possible mix-up: One square meter is 100 square centimeters.

Convert both dimensions: 100 × 100 = 10000.

Explain it to yourself

How can units help you detect that a conversion factor was inverted?

Preview the eight practice prompts
  1. At a constant rate, 6 pens cost $15. What do 10 pens cost in dollars?
  2. Convert 2.35 meters to centimeters.
  3. Convert 4500 grams to kilograms.
  4. Convert 1.5 hours to minutes.
  5. Convert 0.08 square meters to square centimeters.
  6. A cost rule is C = 4n + 7. Is C directly proportional to n? Enter yes or no.
  7. A map uses 1 cm for 2.5 km. Two landmarks are 7.2 cm apart on the map. How far apart are they in kilometers? New context
  8. A rectangular panel measures 40 cm by 75 cm. What is its area in square meters? New context
Open stage 7.2 in the student workspace →

STAGE 7.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Speed, Work & Combined Rates

Useful preparation: Proportions & Unit Conversions · Combining & Scaling Fractions

Goal: Relate amounts, elapsed time and rates with consistent units.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Rate describes amount per unit time

Distance equals speed times time for constant speed. Average speed for a trip is total distance divided by total elapsed time.

d = vt; average speed = total distance / total time

Rates can combine

Independent workers completing the same task contribute fractions of a task per unit time. Add compatible rates, not the completion times.

Combined rate = r₁ + r₂

Relative speed tracks a changing gap

For motion in the same direction the gap changes at the speed difference. For motion toward one another it closes at the speed sum.

Meeting time = initial separation / closing speed
WORKED EXAMPLE 1

A rider goes 12 km in 30 minutes and 18 km in 90 minutes. Find average speed.

  1. The total is 30 km over 2 hours.
  2. Divide distance by total time.

15 km/h

WORKED EXAMPLE 2

Two printers finish a job alone in 6 hours and 3 hours. How long together?

  1. Their rates are 1/6 and 1/3 job per hour.
  2. The total rate is 1/2 job per hour.

2 hours

Common pitfalls

Possible mix-up: Average two speeds without checking time weights.

Use total distance divided by total time.

Possible mix-up: Add completion times to get combined work time.

Add work rates, then invert the combined rate.

Explain it to yourself

Why can equal-distance legs require a different average than equal-time legs?

Preview the eight practice prompts
  1. A walker moves at 4.5 km/h for 2 hours. How many kilometers are covered?
  2. A train covers 210 km in 3.5 hours. Find its average speed in km/h.
  3. How many hours does a 135 km trip take at 45 km/h?
  4. A cyclist rides 10 km in 1 hour, then 30 km in 2 hours. Find the average speed for the whole trip in km/h.
  5. One pump fills a tank in 4 hours and another in 12 hours. Working together, how many hours do they need?
  6. Two runners approach each other from 6 km apart at 8 km/h and 10 km/h. How many hours until they meet?
  7. A conveyor loads 18 boxes per minute while another belt removes 11 per minute. Starting empty, how many boxes accumulate after 7 minutes? New context
  8. A cyclist has a 12 km head start. A car follows the same route at 50 km/h while the cyclist continues at 20 km/h. How many hours after the car starts until it catches up? New context
Open stage 7.3 in the student workspace →
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