Ray's New Practical Arithmetic · Arts. 230–232 · Unit 23: Partnership, Equation of Payments and Average

108. Equation of payments and average

Goal: Your child finds the average time for paying several debts at once, and the average price of a mixture.

⏱ About 40 minutes · 6 steps

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You'll need

  • Paper and pencil, or a slate or small whiteboard

What's in the book

Page 295 defines Equation of Payments (Art. 230) with a worked example and rule. Page 296 has problems and dated purchases (Art. 231). Page 297 covers Average (Art. 232) with a worked example, rule and problems. Answers are printed.

The lesson, step by step

  1. Warm-up

    Ask: You owe $2 in 3 months and $4 in 6 months. If you pay it all at once, should it be closer to 3 months or to 6? Why?

  2. Equation of payments

    Equation of payments is the method of finding the mean or average time of making two or more payments, due at different times.

    A owes B $2, due in 3 mo., and $4, due in 6 mo.: at what period can both sums be paid so that neither party will be the loser?

    2 × 3 = 6 and 4 × 6 = 24. 30 ÷ 6 = 5 months.

    1. Multiply each payment by the time to elapse till it becomes due. 2. Divide the sum of the products by the sum of the payments; the quotient will be the equated time.

  3. Dated purchases

    In finding the Average or Mean time for the payment of several sums due at different times, any date may be taken from which to reckon the time.

    Ray's example: goods bought May 1 ($100) and June 15 ($200) on 60 days' credit. Count from May 1: 100 × 60 + 200 × 105 = 27000; ÷ 300 = 90 days, so July 30.

  4. Average of a mixture

    Average is the method of finding the mean or average price of a mixture, when the ingredients composing it, and their prices, are known.

    I mix 4 pounds of tea, worth 40 ct. a lb., with 6 lb., worth 50 ct. a lb.: what is 1 lb. of the mixture worth?

    $1.60 + $3.00 = $4.60 for 10 lb., so 46 ct. a pound.

  5. Guided practice

    Do these together:

    A owes B $300; 1 third due in 6 mo.; 1 fourth in 8 mo.; the remainder in 12 mo.: what is the average time of payment?

    I have 30 sheep: 10 are worth $3 each; 12, $4 each; the rest, $9 each: find the average value.

  6. Independent practice and check

    Your child works the remaining problems on pages 296–297. The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.

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