Methods and worked examples: Proportional Reasoning-2

2.1 Pie charts: a circle divided in a ratio

A pie chart shows the parts of a whole as slices of a circle. The whole circle is 360 degrees, and each slice's angle is proportional to the quantity it shows.

The chapter's example has grades A to E scored by 40 students: A 12, B 10, C 8, D 6 and E 4. The ratio 12 : 10 : 8 : 6 : 4 can be simplified by dividing every term by the HCF 2, giving 6 : 5 : 4 : 3 : 2. The sum of the simplified terms is 6 + 5 + 4 + 3 + 2 = 20, so one part is 360/20 = 18 degrees.

The angles add to 108 + 90 + 72 + 54 + 36 = 360 degrees, which confirms the calculation.

To draw the chart, draw a circle and mark a radius AB. Measure the first angle anti-clockwise from AB and draw the next radius AC. Continue from each new radius: measure 90 degrees from AC for grade B, 72 degrees from AD for grade C, and so on. Label and colour the slices at the end.

Worked example 1 (school sports survey). In a survey of 200 students, 80 chose cricket, 60 chose football, 40 chose hockey and 20 chose other sports. How large is each slice?

Step 1. The counts add to 200 students, which is the whole. Step 2. Cricket = 80/200 × 360 = 144 degrees. Step 3. Football = 60/200 × 360 = 108 degrees, hockey = 40/200 × 360 = 72 degrees and other = 20/200 × 360 = 36 degrees. Step 4. Check: 144 + 108 + 72 + 36 = 360 degrees.

Worked example 2 (a family budget). A family spends 40% on food, 25% on rent, 20% on education and 15% on savings. Convert the percentages to pie angles by multiplying by 360/100 = 3.6.

Step 1. Food = 40 × 3.6 = 144 degrees. Step 2. Rent = 25 × 3.6 = 90 degrees. Step 3. Education = 20 × 3.6 = 72 degrees. Step 4. Savings = 15 × 3.6 = 54 degrees. Total 360 degrees.

2.2 Inverse proportion

Sometimes one quantity increases while the other decreases by the same factor. The chapter's table shows the trip from Lucknow to Kanpur, written here from the fastest case to the slowest:

Cycling is three times as fast as walking, because 15 divided by 5 is 3, and the trip time falls by the same factor, since 18 divided by 6 is 3. In every row the product speed × time = 90 km, the distance between the two cities. Quantities like these are in inverse proportion.

Two quantities vary inversely when their product stays fixed: xy = k. Writing two pairs as (x1, y1) and (x2, y2), the same constant means x1 y1 = x2 y2. The constant here is the fixed distance, work or quantity of water.

Worked example 3 (workers and bricks). A working day is enough for 5 workers to shift 4,500 bricks. How many workers will shift 18,000 bricks in one day?

Step 1. More bricks need more workers, so this is direct proportion. Step 2. Set 4500 : 18000 :: 5 : x. Step 3. x = 18000 × 5 / 4500 = 20. So 20 workers are needed.

Worked example 4 (pumps). 2 pumps fill a tank in 18 hours. How long will 4 pumps of the same kind take?

Step 1. Doubling the pumps halves the time, so the quantities are inversely proportional. Step 2. 2 × 18 = 4 × x. Step 3. x = 36/4 = 9 hours.

Worked example 5 (food provisions). Food in a school store lasts 15 days for 80 students. Twenty additional students join. Find the new number of days.

Step 1. With more students the food lasts fewer days, so this is inverse. Step 2. 80 × 15 = 100 × x. Step 3. x = 1200/100 = 12 days.

Worked example 6 (working together). Ram cuts a quantity of vegetables in 1 hour and Shyam takes 1.5 hours. How long do they take together?

Step 1. In one hour Ram does 1 unit of work and Shyam does 1/1.5 = 2/3 of the work. Step 2. Together they do 1 + 2/3 = 5/3 units in one hour. Step 3. The time for 1 unit is 1/(5/3) = 3/5 hour, which is 36 minutes.

Note carefully that time and work are directly proportional here: the more units of work, the more time needed. The rates, not the times, are added.

2.3 Direct or inverse: how to decide

Ask what happens to the second quantity when the first is multiplied by a factor.

• If both quantities are multiplied by the same factor, they are directly proportional, and the quotient y/x stays constant.
• If one is multiplied by a factor and the other is divided by that same factor, they are inversely proportional, and the product xy stays constant.

Examples: pencils and their cost are direct; workers and the days needed for a fixed job are inverse; cloth length and price are direct; a cyclist's speed and the time for a route of fixed length are inverse. A useful check is to multiply or divide a pair of values and see whether the product or the quotient stays fixed.