Ray's New Higher Arithmetic · Arts. 234–237 · Unit 14: Percentage
63. Percentage: base, rate and percentage
Goal: Your student uses the words base, rate, percentage, amount and difference, and finds a percentage of a number and what per cent one number is of another.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Percentage defined, per cent, its sign, the elements (base, rate, percentage, amount, difference) and their letters, Case I (base and rate given, find the percentage) and Case II (base and percentage given, find the rate), with principles, formulas, rules, worked examples and many examples for practice.
The lesson, step by step
Warm-up
Mental: 10% of 80; 50% of 36; 25% of 200.
The idea
Read:
Percentage is a term applied to all calculations in which 100 is the basis of comparison.
Per cent comes from the Latin per centum, "by the hundred."
The Base is the number on which the Percentage is estimated.
The Rate is the number of hundredths to be taken.
The Amount is the Base plus the Percentage.
Case I
Read:
Multiply the base by the rate expressed decimally; the product is the percentage.
Work the book's example: 160 sheep, sell 35%: 160 × .35 = 56 sheep. Or: 35% = 7/20, and 7/20 of 160 = 56.
Guided practice
Together:
Find 62 1/2% of 1664 men.
Find 16 2/3% of 1932 hogs.
(1040 men; 322 hogs. Use 5/8 and 1/6.)
Case II
Read:
The rate equals the number of hundredths that the percentage is of the base.
Divide the percentage by the base; the quotient is the rate;
Work the book's example: What per cent of 45 is 9? 9 ÷ 45 = .20 = 20%.
Independent practice
Your student works a selection from each case. Try:
How many % of his time does a man rest, who sleeps 7 hr. out of every 24?
(29⅙%.)
Check
Answers are printed for nearly all. Case I, Example 26 has none; it is below.
Wrap-up
Ask: Is 100% of a number the same as the number? Is 150% more or less than the number?
Afterwards
Two sittings: Case I one day, Case II the next.
Answers to the book's problems
Case I, Example 26 ("What number increased by 20% of 3.5, diminished by 12½% of 9.6, gives 3½?"): 20% of 3.5 = .7; 12½% of 9.6 = 1.2; the number + .7 − 1.2 = 3.5, so the number is 4.
Worked out for this site from the scan. If a number in your copy differs, trust the book.