Ray's New Elementary Algebra · Arts. 164–168 · Unit 5: Chapter V: Supplement to simple equations
33. Formulas and general rules
Goal: Your student solves a problem with letters in place of numbers and reads the answer as a rule in words.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Literal equations, formulas and rules; general solutions for a number divided by two numbers, two numbers from their sum and difference, two workers together, and dividing gain or loss in a partnership, each with short exercises. Answers are printed.
The lesson, step by step
Warm-up
Find a number which, divided by 3 and by 5, gives quotients that add to 16. (30.)
Formula and rule
Read Art. 164.
A Formula is the answer to a problem, when the known quantities are represented by letters.
A Rule is a formula expressed in ordinary language.
A general solution
Read Art. 165, example 3: x/a + x/b = c gives x = abc/(a + b).
Multiply together the three given numbers, and divide the product by the sum of the divisors; the result will be the required number.
Test it on the warm-up: 3 × 5 × 16 ÷ 8 = 30.
Sum and difference
Read Art. 166.
1. To find the greater, add half the difference to half the sum.
2. To find the less, subtract half the difference from half the sum.
Your student does exercise 3 (sum 200, difference 50; Ans. 125, 75).
Working together
Read Art. 167.
Divide the product of the numbers expressing the time in which each can perform the work, by their sum; the quotient will be the time in which they can jointly perform it.
Your student does exercise 3: pipes taking 6 and 9 hours fill the cistern in 54 ÷ 15 = 3 3/5 hours.
Partnership
Read Art. 168 and work exercise 3 together: stocks of $200, $300 and $700 are as 2, 3 and 7; a $60 loss divided by 12 is $5, so the shares are $10, $15 and $35.
Independent practice
The remaining exercises of Arts. 165 to 168.
Wrap-up
Ask: Why is a formula more useful than one answer?
Tip
Testing a formula on a problem already solved with numbers is the best way to trust it.
Answers to the book's problems
The book prints the answer after each exercise.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.