Direct and Inverse Proportion, Scale Plans and Unit Rates

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Opens on Chapter 23’s example: 3 bags of gravel cover 24 square feet, and Maeve’s patio base is 126 square feet; 3 people take 8 hours to lay it.

1. Direct: y = kx

Amount bags

Covers sq ft

To cover sq ft

2. Inverse: x × y = k

People

Hours

With this many people

3. A scale plan

1 inch on paper = feet

Length on paper (in)

Width on paper (in)

4. Test a table

x values

y values

Predict y at x =

5. Proportional or not?

Setup fee

Rate per item

Items

and

Extras

A recipe: cups

for people

scaled to people

Flip a rate: mph

Enter your numbers, then press Work it out.

How it works

Direct proportion has a constant ratio y ÷ x = k, so y = kx and a graph through zero; find k from one pair and multiply. Inverse proportion has a constant product x × y = k: more workers, fewer hours, so the new time is k ÷ the new number (multiplying by 4/3 instead goes the wrong way). On a scale plan, lengths scale by the scale factor but areas by its square: 1 inch to 4 feet makes each square inch 16 square feet. A formula with a fixed fee isn’t proportional: doubling the items doesn’t double the cost, which is why the cost per item falls. A rate flips: 45 miles an hour is 60 ÷ 45 minutes a mile.

From the book, Chapter 23. 3 bags covering 24 sq ft is 8 sq ft a bag, so the 126 sq ft base takes 15.75 bags: buy 16. A patio of 24 person-hours takes 4 people 6 hours and 6 people 4 (not 10.7). 3.5 by 2.25 inches at 1 inch to 4 feet is 14 by 9 ft, 126 sq ft: 7.875 sq in × 16, a scale of 1 : 48. The pump table multiplies to 1,200, so 50 gallons a minute takes 24 minutes. Sanjay’s 100 sheets cost $52 and 200 cost $92, not $104 ($0.52 then $0.46 a sheet). Rice for 10 is 3 3/4 cups; 45 mph is 1 1/3 minutes a mile. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. The algebra is done with exact fractions, so 1/3 stays 1/3; decimals are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.