Ray's New Higher Arithmetic · Arts. 362–363 · Unit 24: Alligation

106. Alligation alternate: how much of each?

Goal: Your student finds in what proportions to mix things of given prices to get a wanted average price.

⏱ About 40 minutes · 6 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

Alligation alternate, Case I (no quantity limited): two worked problems explained by gains and losses, the balancing rule, a note and remark, and examples. Answers are printed; other correct answers are possible.

The lesson, step by step

  1. Warm-up

    Ask: Sugar at 9 cents and at 5 cents is mixed to sell at 6 cents. Selling the 9-cent kind at 6 loses 3 cents a pound. Selling the 5-cent kind at 6 gains 1 cent. How many pounds of the cheap kind balance one pound of the dear kind? (3.)

  2. The idea

    Read:

    Alligation Alternate is the process of finding the proportional quantities at given particular prices or values in a required combination of given average value.

    Read the principle:

    each losing sale must be balanced by one which gains.

  3. Work the book's example

    What relative numbers of hogs, at $3, $5, $10 per head, can be bought at an average value of $7 per head?

    The $3 and $5 hogs gain 4 and 2; the $10 hog loses 3. Balance each gain against the loss, as the book shows. One answer is 1, 1, 2.

  4. The rule

    Read the first step of the book's rule:

    Write the particular values or prices in order, in a column, having the smallest at the head; write the average value in a middle position at the left and separated from the others by a vertical line.

    Then follow the book's balance columns.

  5. Guided practice

    What relative quantities of tea, worth 25, 27, 30, 32, and 45 ct. per lb., must be taken for a mixture worth 28 ct. per lb.?

    Check any answer by alligation medial. Read:

    The proof of Alligation Alternate is the process of Alligation Medial.

  6. Independent practice and wrap-up

    Your student works a few more. If their answer differs from the book's, check it by medial; it may still be right. Ask: Why can there be more than one answer?

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