🔢 MathGrade 6Lesson 9
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Write and Interpret Ratios

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Learning Objectives

  • 1Write a ratio comparing two quantities in three notations: a:b, a to b, and a/b
  • 2Simplify a ratio to lowest terms by dividing both terms by their greatest common factor
  • 3Generate equivalent ratios by multiplying or dividing both terms by the same nonzero number
  • 4Distinguish part-to-part ratios from part-to-whole ratios in everyday situations
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Mini Lesson

A ratio is a way to compare two quantities by how many of one there are for each amount of the other. Ratios show up in recipes (3 cups flour to 2 cups water — about 720 mL to 480 mL), sports (15 wins to 5 losses), maps (1 cm represents 50 km), and money (2 dollars per 5 candies). The order of the two quantities matters — a 3:5 ratio is not the same comparison as 5:3.

Three Ways to Write the Same Ratio

  • Colon notation: 3:5 (read "three to five").
  • Word notation: 3 to 5.
  • Fraction notation: 3/5.
  • All three forms name the SAME comparison and can be swapped freely.
  • In every form, the FIRST number names the first quantity and the SECOND number names the second quantity. Order matters: 3:5 is NOT the same as 5:3.

Simplifying a Ratio to Lowest Terms

  • A ratio is in lowest terms when its two numbers share no common factor greater than 1.
  • To simplify, divide BOTH numbers by their greatest common factor (GCF — see Lesson 5).
  • Example: 6:9 — GCF of 6 and 9 is 3, so 6:9 = (6/3):(9/3) = 2:3.
  • Example: 12:20 — GCF of 12 and 20 is 4, so 12:20 = 3:5.
  • Simplifying never changes the comparison; it just makes the ratio easier to read.

Equivalent Ratios — Same Comparison, Different Numbers

  • Multiply OR divide both numbers by the same nonzero amount to make an equivalent ratio.
  • Multiplying: 2:3 → (2 x 4):(3 x 4) → 8:12. So 2:3 and 8:12 are equivalent.
  • Dividing: 12:20 → (12/4):(20/4) → 3:5. So 12:20 and 3:5 are equivalent.
  • You CANNOT add or subtract the same amount to make an equivalent ratio — you must multiply or divide.
  • A ratio table shows several equivalent ratios in a row: 3:5, 6:10, 9:15, 12:20 are all equivalent.

Part-to-Part vs Part-to-Whole

  • Part-to-part: compare one part of a group to another part. Example — in a club with 12 boys and 15 girls, the boys-to-girls ratio is 12:15 = 4:5.
  • Part-to-whole: compare one part of a group to the entire group. Example — boys to total kids is 12:27 = 4:9.
  • Always read the words carefully to decide which kind of ratio is being asked for.
  • Part-to-whole ratios can also be written as fractions of the total: 12/27 of the club is boys.
Writing a Ratio from a PictureThree blue circles and five red squares illustrate the ratio three to five in three notations: 3:5, 3 to 5, and 3/5.Writing a Ratio from a PictureCirclesSquares35circles to squares3 : 53 to 53 / 5All three notations name the SAME ratio.Order matters: 3:5 is not the same as 5:3.
Three ways to write one ratio — order always matters.
Equivalent Ratios for 3 to 5Table showing equivalent ratios: 3 to 5, 6 to 10, 9 to 15, and 12 to 20, all named by multiplying both terms by the same factor.Equivalent Ratios for 3:5x1x2x3x4Circles36912Squares5101520Multiply both terms by the same number.3:56:109:1512:20All equivalent to 3:5
Multiply both numbers by the same factor for equivalence.
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Tip: when you write a ratio from a word problem, underline the two quantities being compared and write them in the SAME order they appear in the question. "Apples to oranges" means apples first, oranges second — never swap the order.

Vocabulary
ratio
A comparison of two quantities, written as a:b, a to b, or a/b.
equivalent ratio
A ratio that names the same comparison as another, found by multiplying or dividing both terms by the same nonzero number.
simplify
Write a ratio in lowest terms by dividing both numbers by their greatest common factor.
part-to-part
A ratio that compares one part of a group to another part of the same group.
part-to-whole
A ratio that compares one part of a group to the total amount in the whole group.
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