Ray's New Higher Arithmetic · Arts. 218–220 · Unit 11: Compound Denominate Numbers
53. Reduction of compound numbers
Goal: Your student changes a quantity to a lower denomination by multiplying and to a higher one by dividing, step by step.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Reduction defined, with its two principles (descending by multiplication, ascending by division), worked examples (18 bushels to pints, 236 inches to yards, 9.375 acres to square rods, and more), the two rules, a remark, and a note on the metric system.
The lesson, step by step
Warm-up
Mental: How many quarts in 3 gallons? How many gallons in 20 quarts?
The idea
Reduction is:
changing them to equivalent numbers of a different denomination.
Reduction from a higher denomination to a lower, is performed by multiplication.
Reduction from a lower denomination to a higher, is performed by division.
Work the book's example
Reduce 18 bushels to pints. 18 × 4 = 72 pecks; × 8 = 576 quarts; × 2 = 1152 pints. The book's short way: 1 bu. = 64 pt., and 64 × 18 = 1152.
Going up
Read:
Divide the given quantity by that number of its own denomination which makes a unit of the next higher.
Work 236 inches to yards: 236 ÷ 12 = 19 ft. 8 in.; 19 ÷ 3 = 6 yd. 1 ft. So 6 yd. 1 ft. 8 in., or 6 5/9 yd.
Mixed denominations
To reduce 3 yd. 2 ft. 5 in. to inches: 3 × 3 = 9, + 2 = 11 ft.; 11 × 12 = 132, + 5 = 137 in. The book's remark:
Abstract factors can not produce a concrete result
is why we name each step's unit.
Metric
Read:
In the Metric System the operations are performed by removing the point to the right or to the left.
The book's example: 1238.73 hektograms = 123873 grams.
Wrap-up
Ask: Which gives more units, reducing descending or ascending? Read the book's remark:
Reduction Descending diminishes the size, and, therefore, increases the number of units given