Ray's New Higher Arithmetic · Arts. 364–365 · Unit 24: Alligation
107. Alligation alternate: limited quantities and totals
Goal: Your student solves mixture problems where one quantity, or the whole amount, is fixed.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Case II (one quantity given) and Case III (the whole amount given), each with a worked problem, a rule, remarks and examples; the topical outline of Alligation follows. Answers are printed.
The lesson, step by step
Warm-up
Review: find one balanced mixture of tea at 30 and 50 cents to sell at 40. (1 and 1.)
Case II
Read:
To proportion the parts, one or more of the quantities, but not the amount of the combination, being given.
The book's example: wheat at $1.20 and $1.40, with 14 bushels at $1.90, to average $1.60. Case I gives the proportions 1, 1, 2. Since the third must be 14, one answer is 7, 7, 14.
Guided practice
How many railroad shares, at 50%, must A buy, who has 80 shares that cost him 72%, in order to reduce his average to 60%?
Case III
Read:
To proportion the parts, the amount of the whole combination being given.
If a man pay $16 for each cow, $3 for each hog, and $2 for each sheep, how many of each kind may he purchase so as to have 100 animals for $600?
The balance columns give 20 animals. Multiply by 5 to make 100: 25 cows, 50 hogs and 25 sheep. Check: $400 + $150 + $50 = $600.
Guided practice
How many bbl. flour, at $8 and $8.50, with 300 bbl. at $7.50, and 800 at $7.80, and 400 at $7.65, will make 2000 bbl. at $7.85 a bbl.?
Independent practice
Your student tries more examples, such as the gold ring problem. Check each answer by alligation medial.
Wrap-up
Read the topical outline of Alligation together and name the three cases of alligation alternate.