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Unit 3 · Standard 6.AT.3c

Ratio Reasoning: Convert Measurements within the Same System

Lesson 3-6 · Learn It — how the math works

Why we're learning this

Chef Reyes writes every recipe in the units the supplier uses, but the kitchen's measuring tools are marked in different units of the same system. A stock order lists quarts; the pot is marked in cups. A height rule at the county fair ride is posted in inches; the sign-up sheet lists feet. Your job is to convert between units without changing the amount. The stock order lists 3 quarts, but the pot is marked in cups. The height rule at the ride says 48 inches, but the sign-up sheet lists feet.

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Converting inside one measurement system

Two measurements can only be compared when they are written in the same unit. Converting uses a special ratio — a unit ratio — that says how much of the smaller unit fits into exactly 1 of the larger unit.

Picture itSee what the math looks like before you read about it

Chef Reyes lines up two trays of vinaigrette at the tasting table. Tray A mixes 3 tablespoons of vinegar with 9 tablespoons of oil. Tray B mixes 5 tablespoons of vinegar with 15 tablespoons of oil.
Where you see this in real lifeChef Reyes lines up two trays of vinaigrette at the tasting table. Tray A mixes 3 tablespoons of vinegar with 9 tablespoons of oil. Tray B mixes 5 tablespoons of vinegar with 15 tablespoons of oil.

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram: 4 feet, each 12 inches, make 48 inches.48Inches121 foot121212
  1. A ride requires riders to be at or above 44 inches tall. A child is 4 feet tall. The two measurements use different units, so I cannot compare them yet.
  2. Both units belong to the customary system, and there are 12 inches in one foot. That relationship is the . As a ratio it is 12 : 1 — a , because it compares an amount to exactly 1 foot.
  3. Inches are smaller than feet, so there will be MORE of them: I multiply. 4 × 12 = 48.4 × 12 = 48
  4. The child is 48 inches tall. Now the comparison is fair: 48 inches is greater than 44 inches, so the child can go on the ride.

Watch out — mistakes students make here

  • Dividing converts to the LARGER unit. Cups are smaller than quarts, so the number must grow.
  • The amount of broth does not change, but the NUMBER does, because the unit changed.
  • Converting scales the amount by the unit ratio; it is never an addition.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Diamond wants to drink 4 liters of water. Her bottle is marked in milliliters.
  2. The is 1,000 mL : 1 L. On a , every 1 liter above matches 1,000 milliliters below.
  3. Milliliters are smaller, so multiply: 4 × 1,000 = 4,000. So 4 liters = 4,000 milliliters.4 × 1,000 = 4,000
  4. Her bottle holds 500 milliliters. How many fills?
    Check this step
    Solve ? × 500 = 4,000. Since 8 × 500 = 4,000, she fills the bottle 8 times.8 × 500 = 4,000

On your ownYou do — solve this one by yourself

Before you start, remember:

  • A stockpot recipe calls for 3 quarts of broth, but the pot is marked only in cups.
  • Write the : there are 4 cups in 1 quart, so 4 : 1.
  • Are cups smaller or larger than quarts? Decide whether to multiply or divide.
  • 3 quarts to cups, then say the answer as a full sentence with its unit.

Solve: Diamond needs 4 liters of water. Her bottle holds 500 milliliters. How many times must she fill it?

My answer

Stuck? Open one hint at a time.

Hint 1

Get both amounts into the same unit first.

Hint 2

4 liters is 4,000 milliliters.

Hint 3

What times 500 equals 4,000?

Check my answer
✅ Answer: 8 times
Convert first: 4 L = 4,000 mL. Then solve ? × 500 = 4,000, giving 8 fills.

Turn & TalkSay it out loud with a partner

Chef Reyes's recipe serves 8 people, but the banquet has 120 guests. How many times bigger is 120 than 8, and how does that scale factor help you adjust the tomatoes and mozzarella?

Try starting with:

  • 120 guests is ___ times bigger than 8 because 120 divided by 8 equals ___.120 invitados es ___ veces más que 8 porque 120 entre 8 es ___.
  • So I scale by ___, which keeps the recipe in ___.Entonces escalo por ___, lo que mantiene la receta en ___.
Use these wordsproportionscaleequivalentunit rateratio