Ray's New Higher Arithmetic · Arts. 323–326 · Unit 21: Percentage with Time: Equation of Payments and Settlement of Accounts
93. Equation of payments
Goal: Your student finds the single date on which several debts, due at different times, can fairly be paid at once.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
- A calendar
What's in the book
Section VI, Equation of Payments: definitions (equated time, term of credit, average term of credit), worked problems, the rule for simple equations, compound equations with debits and credits, the product method, and examples. Answers are printed.
The lesson, step by step
Warm-up
Ask: You owe $100 due in 1 month and $100 due in 3 months. If you pay $200 all at once, what day is fair? (In 2 months.)
The idea
Read:
Equation of Payments is the process of finding the time when two or more debts, due at different times, may be paid without loss to either debtor or creditor.
The time of payment is called the Equated Time.
The Term of Credit is the time to elapse before a debt is due.
Work the book's example
A buys $250 on 3 months' credit, $800 on 2 months and $1000 on 4 months. Multiply each debt by its months: 750 + 1600 + 4000 = 6350. Divide by the total debt, $2050: 3.1 months, or 3 mo. 3 da.
Read the rule:
Multiply each debt by its term of credit, and divide the sum of the products by the sum of the debts; the quotient will be the equated time.
With dates
Work the book's dated example: invoices of Jan. 7, 11, 13, 20 and 28. Count days from Jan. 7, multiply, add, and divide. The answer is about 12 days after Jan. 7, so the note is dated Jan. 19.
Accounts with both sides
Read:
If the account has credits as well as debits, it is called a compound equation, and may be averaged on the same principle as the simple equation.
Look at the book's worked account. The balance is the difference of the items, and the time comes from the difference of the products.
Practice
Your student works the two-debt problems first, such as:
I owe $912, due Oct. 16, and $500, due Dec. 20. If I pay the first, Oct. 1, 15 days before due, when should I pay the last?
Then one account from the page.
Check and wrap-up
Check the printed answers. Ask: If you pay one debt early, should you get to pay another one late? Why is that fair?
Afterwards
Two sittings: simple equations one day, compound accounts the next.