Ratio

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What a Ratio Is

  • A ratio compares how many times one amount contains another.
  • If there are 8 oranges and 6 lemons, the ratio of oranges to lemons is 8:6, which simplifies to 4:3.
  • The ratio of lemons to oranges is 6:8, or 3:4 — the order matters.
  • The ratio of oranges to the total fruit is 8:14, or 4:7.
  • Ratios can compare counts of objects, or measurements of length, weight, time and more.
  • In most contexts, both numbers in a ratio are positive.

Notation and Terms

  • A ratio can be written as A:B, as "A to B", or as the fraction \frac{A}{B}.
  • In A:B, A is the antecedent and B is the consequent; together they are the terms of the ratio.
  • The fraction form \frac{A}{B} can be written as a simple fraction, a decimal, or a percentage.
  • Equal quotients give equal ratios: if \frac{A}{B} = \frac{C}{D}, then A:B = C:D.
  • A statement that two ratios are equal, such as A:B = C:D, is called a proportion.
  • In a proportion A:B = C:D, A and D are the extremes, and B and C are the means.
  • Ratios can have three or more terms, e.g. cement : sand : gravel = 1:2:4.

Simplifying Ratios

  • To simplify a ratio, divide all terms by their highest common factor.
  • 8:6 simplifies to 4:3 because both terms divide by 2.
  • Simplified ratios keep the same relationship: 4:3 and 8:6 are equivalent.
  • When units differ, convert to the same unit first, then simplify.
  • Example: 30 minutes to 2 hours becomes 30:120, which simplifies to 1:4.

Writing a Ratio as 1 : n

  • To write A:B in the form 1:n, divide both terms by A.
  • Example: 5:20 becomes 1:4 because 5 ÷ 5 = 1 and 20 ÷ 5 = 4.
  • This form is useful for comparing sizes directly, e.g. a scale of 1:50.
  • If the first term is not 1, you can still divide both terms by the first term to make it 1.

Sharing in a Given Ratio

  • To share an amount in the ratio A:B, first find the total number of parts: A + B.
  • One part = total amount ÷ (A + B).
  • Then multiply one part by A and by B to get each share.
  • Example: share £60 in the ratio 2:3. Total parts = 5, so one part = £12. Shares are £24 and £36.
  • For three-term ratios, add all three terms to find the total number of parts.

Finding the Whole or Another Part

  • If you know one part and its ratio value, you can find the other parts.
  • Example: if 3 parts = 15, then 1 part = 5.
  • Multiply 1 part by the other ratio numbers to find the missing amounts.
  • To find the whole, add all the parts together after finding the value of one part.
  • Example: ratio 2:5, and the 2-part is 10. Then 1 part = 5, the 5-part = 25, and the whole = 35.

Linking Ratios to Fractions

  • In the ratio A:B, the first part is \frac{A}{A+B} of the whole.
  • The second part is \frac{B}{A+B} of the whole.
  • Example: in the ratio 3:2, the first part is \frac{3}{5} and the second is \frac{2}{5} of the total.
  • This link helps when solving problems that mix fractions and ratios.

Combining Ratios

  • To combine two ratios that share a common term, make that term the same in both.
  • Example: if A:B = 2:3 and B:C = 4:5, make B the same. Multiply the first ratio by 4 and the second by 3 to get A:B = 8:12 and B:C = 12:15.
  • Then A:B:C = 8:12:15.
  • Once combined, you can use the three-part ratio to share amounts or solve problems.

Best-Buy and Recipe Problems

  • A best buy compares cost per unit to find the better value.
  • To compare, work out the price for the same quantity, e.g. cost per 100 g or per item.
  • Example: 500 g for £2.50 is 50p per 100 g; 750 g for £3.60 is 48p per 100 g, so the second is better value.
  • In recipe problems, scale all ingredients by the same factor to keep the ratio the same.
  • If a recipe for 4 people uses 200 g flour, for 6 people use 200 \times \frac{6}{4} = 300 g.
  • Keep the ratio of ingredients unchanged when changing the number of servings.

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 61 câu hỏi. Đăng ký để xem tất cả.

  1. 1.In the ratio A:B, what is A called?

    Easy
    • AThe antecedent
    • BThe consequent
    • CThe extreme
    • DThe mean

  2. 2.In the proportion A:B = C:D, which pair are the extremes?

    Easy
    • AA and D
    • BB and C
    • CA and B
    • DC and D

  3. 3.A fruit bowl contains 8 oranges and 6 lemons. What is the ratio of oranges to the total amount of fruit, in simplest form?

    Easy
    • A4:7
    • B4:3
    • C3:4
    • D7:4

  4. 4.Which of the following are valid ways to express the ratio of 3 to 4? (select all that apply)

    Medium
    • A3:4
    • B3 to 4
    • C\frac{3}{4}
    • D4:3
    • E3 is to 4

  5. 5.A statement expressing the equality of two ratios is called a proportion.

    Easy

    True or false?

  6. 6.The ratio 6:8 is proportional to the ratio 3:4.

    Easy

    True or false?

  7. 7.Match each ratio term to its meaning in the proportion A:B = C:D.

    Medium
    • A
    • D
    • B
    • C
    • first extreme
    • second extreme
    • first mean
    • second mean

  8. 8.A 'two by four' piece of wood has thickness : width : length = 2:4:10. Which statement is true?

    Medium
    • AThe ratio of thickness to width is 1:2.
    • BThe ratio of width to length is 1:2.
    • CThe ratio of thickness to length is 1:4.
    • DThe ratio of width to thickness is 2:1.

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