Ray's New Elementary Algebra · Art. 156 (problems 9–40) · Unit 4: Chapter IV: Simple equations
28. Word problems: work, mixtures and shares
Goal: Your student solves work-rate, mixture and sharing problems with one unknown.
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
Worked problems 9 to 15 (two pipes filling a cistern, three men at work, sugar mixture, the laborer, the hare and greyhound, the watch hands, spent money), the advice on numbers in a given ratio, and exercises 16 to 40 with printed answers.
The lesson, step by step
Warm-up
Ask: If a pipe fills a tank in 40 minutes, how much of the tank does it fill in 1 minute? (1/40.)
Work problems
Read problem 9. Say: Add what each does in one minute: 1/40 + 1/30 = 1/x. Multiply by 120x: 3x + 4x = 120, so x = 17 1/7 minutes.
Then read problem 10; finish its last line together: 59x = 120, so x = 120/59, or 2 2/59 days.
Mixtures
Read problem 11 (sugar at 5 and 9 cents making 100 pounds at 6 cents). Say: The value of the parts equals the value of the mixture: 5x + 9(100 − x) = 600.
Numbers in a ratio
Read problem 23 and the direction after it:
When two or more unknown quantities in any problem are to each other in a given ratio, assume each of them a multiple of an unknown quantity, so that they shall have to each other the given ratio.
So for 2 to 3, let the parts be 2x and 3x.
Guided practice
Your student does these while you watch:
16. Divide 42 cents between A and B, giving to B twice as many as to A. Ans. A 14, B 28.
24. The sum of two numbers is 60, and the less is to the greater as 5 to 7 ; what are the numbers ? Ans. 25 and 35.
Independent practice
Choose eight to ten of exercises 17 to 40, mixing kinds. Read problems 12 to 15 together if time allows.
Wrap-up
Ask: In problem 12, why did Ray change the $3 to 300 cents?
Tip
Many of these problems can be acted out with coins or a drawing. Do that whenever your student is stuck.
Answers to the book's problems
The book prints the answers to exercises 16 to 40. Worked problem 10 stops at 59x = 120: the answer is x = 120/59 = 2 2/59 days.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.