Solving problems with ratios

Kesang made 6 glasses of lemonade with 10 spoons of sugar. Her father asks for 18 more glasses at the same sweetness. How much sugar?

Sweetness stays the same only if the ratio of glasses to spoons stays proportional. So: 6 : 10 :: 18 : ?

Find the factor on the first term: 18 ÷ 6 = 3. The second term must change by the same factor: 10 × 3 = 30. So 6 : 10 :: 18 : 30, and she needs 30 spoons.

Next, a worry about a wall. Nitin built the 60 ft side using 3 bags of cement; Hari built the 40 ft side using only 2. Nitin fears Hari's wall is weaker.

Compare cement per length. Nitin: 60 : 3, which reduces to 20 : 1. Hari: 40 : 2, which also reduces to 20 : 1. Proportional: one bag for every 20 ft, so the walls are equally strong, and Nitin can stop worrying.

Now a trap worth remembering. When Neelima was 3, her mother was 10 times her age: 30. The ratio of their ages was 3 : 30, that is 1 : 10.

Nine years later Neelima is 12 and her mother is 39. The ratio is now 12 : 39, which reduces to 4 : 13: not 1 : 10 at all. Both ages grew by the same amount (the 27-year gap between them never changes), and so the ratio changed. Her mother doesn't age 10 years for every year Neelima does! Adding or subtracting the same number from both terms does not keep a ratio proportional. Only multiplying does.

Finally, filter coffee. Manjunath mixes 15 mL of decoction with 35 mL of milk for a regular cup. For a stronger cup he uses 20 mL with 30 mL; for a lighter one, 10 mL with 40 mL.

All three cups are 50 mL: what changes is the ratio. 20 : 30 has more decoction per millilitre of milk than 15 : 35, so it's stronger; 10 : 40 has less, so it's lighter.