Lesson 9 of 10 · 7 min
Word problems: ranges, temperatures and mixtures
NCERT §5.3
At the end of the day Riya has 12 litres of juice concentrate that is 40% fruit. Adding water makes it go further, but the drink must stay tasty: more than 20% fruit and less than 25%. How much water can she add?
The lesson in notes
In short
When a quantity must lie in a range and another quantity depends on it through a formula, write a double inequality and solve it.
Temperature: a cooler kept between 4 °C and 10 °C is, with F = (9/5)C + 32, between 39.2 °F and 50 °F.
Reverse conversion: with C = (5/9)(F − 32), a range 50 < F < 59 gives (5/9)(18) < C < (5/9)(27), that is 10 < C < 15.
Mixture: 12 litres of juice concentrate are 40% fruit, so they hold 4.8 litres of fruit. Add x litres of water. For the fruit share to be more than 20% but less than 25%: 0.20(12 + x) < 4.8 < 0.25(12 + x).
Solving each side: 12 + x < 24 gives x < 12, and 12 + x > 19.2 gives x > 7.2. So between 7.2 and 12 litres of water, not including either end. Check: at x = 7.2 the share is 4.8/19.2 = 25%, and at x = 12 it is 4.8/24 = 20%.
Adding more water lowers the share, so the larger share (25%) matches the smaller amount of water. Checking the ends catches a range written backwards.