Ray's New Higher Arithmetic · Arts. 238–241 · Unit 14: Percentage
64. Finding the base
Goal: Your student finds the base from the percentage and rate, and from the amount or difference and rate.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Case III (rate and percentage given, find the base), Case IV (rate and amount or difference given, find the base), many examples for practice with printed answers, additional formulas, and the two classes of applications of percentage. A topical outline of applications closes the chapter.
The lesson, step by step
Warm-up
Mental: 10 is 10% of what? 5 is 50% of what?
Case III
Read:
The base bears the same ratio to the percentage that 100% does to the rate.
Divide the percentage by the rate expressed decimally; the quotient is the base.
Work the book's example: 95 is 5% of what number? 95 ÷ .05 = 1900.
Guided practice
Together:
$3.80 is 5% of what sum?
$189.80 is 104% of what?
($76; $182.50.)
Case IV
Read:
The base is equal to the amount divided by 1 plus the rate
Work the book's examples: A rents a house for $377, an advance of 16% on last year: $377 ÷ 1.16 = $325. John has $136, which is 20% less than Joseph's: $136 ÷ .80 = $170.
A common trap
Ask: If a price rises 20% and then falls 20%, is it back where it started? (No: 100 → 120 → 96.) This is why Case IV divides instead of subtracting.
Independent practice
Your student works a selection, including:
2576 bu. is 60% less than what?
(6440 bu.)
Formulas and applications
Read:
By definition, A = B + P ; also, D = B — P.
The book then divides the applications of percentage into those where time does not matter (profit and loss, stocks, commission, insurance, taxes...) and those where it does (interest, discount...).
Wrap-up
Look over the topical outline. Ask: Which of these applications have you seen in real life?
Afterwards
Two sittings: Case III one day, Case IV and the extra examples the next.