Ray's New Higher Arithmetic · Arts. 360–361 · Unit 24: Alligation

105. Alligation medial: the average of a mixture

Goal: Your student finds the average price or strength of a mixture.

⏱ About 35 minutes · 7 steps

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You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil

What's in the book

Chapter XVIII: alligation and its two kinds, alligation medial with a worked sugar problem, a rule, and examples (tea, hogs, gold, alcohol, specific gravity). Answers are printed.

The lesson, step by step

  1. Warm-up

    Ask: Mix 1 cup of juice worth 10 cents with 1 cup worth 20 cents. What is a cup of the mix worth? (15 cents.)

  2. The idea

    Read:

    Alligation is the process of taking quantities of different values in a combination of average value.

    Alligation Medial is the process of finding the mean or average value of two or more things of different given values.

  3. Work the book's example

    If 3 lb. of sugar, at 5 ct. a lb., and 2 lb., at 4½ ct. a lb., be mixed with 9 lb., at 6 ct. a lb., what per lb. is the mixture worth?

    15 ct. + 9 ct. + 54 ct. = 78 ct. for 14 lb. 78 ÷ 14 = 5 4/7 ct. a lb.

  4. The rule

    Read:

    Find the values of the definite parts, and divide the sum of the values by the sum of the parts.

  5. Guided practice

    Find the average price of 6 lb. tea, at 80 ct.; 15 lb., at 50 ct.; 5 lb., at 60 ct.; 9 lb., at 40 ct.

    What per cent of alcohol in a mixture of 9 gal., 86% strong; 12 gal., 92% strong; 10 gal., 95% strong; and 11 gal., 98% strong?

  6. Old measures

    The gold problems use carats (24 carats is pure gold) and pennyweights (pwt.; 20 pwt. to the troy ounce). The specific gravity problems compare a metal's weight with the same bulk of water.

  7. Independent practice and wrap-up

    Your student works the other examples and checks the printed answers. Ask: Can the average ever be higher than the dearest item? (No.)

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