Proportionality Recap and Maps

3.1 Proportionality — A Quick Recap

In an earlier chapter, we studied proportional relationships^? between quantities. We used the ratio notation to represent such relationships. When two or more related quantities change by the same factor, we call that relationship a proportional relationship.

For example, idli batter is made by mixing rice and urad dal. The proportion of these two can have several regional variations. One such proportion used is: for 2 cups of rice, add 1 cup of urad dal. We represent this relationship using the ratio 2 : 1.

3.2 Ratios in Maps

Have you noticed that maps have a ratio given, usually in the lower right corner? It usually contains 1 and a very large number, such as 1 : 60,00,000. What does RF 1 : 60,00,000 mean? What does it indicate?

Try to find the distances between the same two pairs of cities with different maps that have different scales (ratios). Do they all give the same geographical distance, approximately? They should — because the ground truth is fixed, only the map scale changes.

3.3 Ratios with More than 2 Terms

Viswanath is experimenting with a spice mix powder. He makes the powder by grinding 8 spoons of coriander seeds, 4 red chillies, 2 spoons of toor dal and 1 spoon of fenugreek (methi) seeds. For his spice mix powder, the ratio of coriander seeds : red chillies : toor dal : fenugreek seeds is:

8 : 4 : 2 : 1

Notice that the ratio has 4 terms. Ratios can have many terms if each of the quantities change by the same factor to maintain the proportional relationship.

Puneet has only 2 red chillies in his kitchen. But he wants to make spice mix powder that tastes the same as Viswanath's. How much of the other ingredients should Puneet use to make his spice mix powder?

Since Puneet has half the number of chillies Viswanath used (2 instead of 4), the quantity of all other ingredients must halve too:

4 : 2 : 1 : 0.5

So Puneet should add 4 spoons of coriander, 2 red chillies, 1 spoon of toor dal, and half a spoon of fenugreek.

Example 1 — Purple Paint Mixture

To make a special shade of purple, paint must be mixed in the ratio, Red : Blue : White = 2 : 3 : 5. If Tumati has 10 litres of white paint, how many litres of red and blue paint should she add to get the same shade of paint?

The ratio 2 : 3 : 5 tells us that the white paint corresponds to 5 parts. If 5 parts is 10 litres, 1 part is 10 ÷ 5 = 2 litres. So:

• Red = 2 × 2 = 4 litres
• Blue = 3 × 2 = 6 litres
• Total purple paint = 4 + 6 + 10 = 20 litres

Example 2 — Concrete Mixture

Cement, sand and gravel are mixed in the ratio 1 : 1.5 : 3 to make concrete. The ratio of the components in the structure needs to be this. For structures that need greater strength like pillars, beams, and roofs, the ratio is 1 : 1.5 : 3, and the construction is also reinforced with steel rods. Using this ratio, if we have 3 bags of cement, how many bags of concrete mixture can we make?

The concrete mixture is in the ratio: Bags of cement : bags of sand : bags of gravel = 1 : 1.5 : 3. If we have 3 bags of cement, we need to multiply the other terms by 3, so the ratio becomes 3 : 4.5 : 9. Total bags = 3 + 4.5 + 9 = 16.5 bags of concrete.

3.4 Dividing a Whole in a Given Ratio

In an earlier chapter, we learned how to divide a whole in a ratio, e.g., 12 in the ratio 2 : 1. To do this, we add the terms (2 + 1 = 3), and divide the whole by this sum (12 ÷ 3 = 4). With the multiply each term by this quotient: \(2 \times 4 = 8\) and \(1 \times 4 = 4\). So 12 divided in the ratio 2 : 1 is 8 : 4.

We can extend this idea to divide a quantity in a concrete mixture. Let us consider 110 units of concrete are needed. How many units of cement, sand and gravel are needed if the ratio is 1 : 1.5 : 3?

For 1 unit of cement, we need to add 1.5 units of sand and 3 units of gravel. In total this gives 1 + 1.5 + 3 = 5.5 units of concrete. So each unit of cement gives us 5.5 units of concrete. To make 110 units, we multiply by 20 (since 110 ÷ 5.5 = 20):

• Cement: 1 × 20 = 20 units
• Sand: 1.5 × 20 = 30 units
• Gravel: 3 × 20 = 60 units

Example 4 — Rangoli Powder

For a particular shade of purple rangoli powder, we mix red, blue and white powder in the ratio 2 : 3 : 5. If you have 50 grams of powder to make, how much of each colour should you use?

Total parts = 2 + 3 + 5 = 10.

• Red = \(50 \times \dfrac{2}{10} = 10\) g
• Blue = \(50 \times \dfrac{3}{10} = 15\) g
• White = \(50 \times \dfrac{5}{10} = 25\) g