25
Applications of direct and inverse proportionality
Textbook, pp. 106–114
GoalSolve problems on direct and inverse proportionality; divide a number in a given ratio.
New words
direct relation · to‘g‘ri bog‘lanishinverse relation · teskari bog‘lanishdividing in a ratio · nisbatda bo‘lishpart / share · bo‘lak
Explanation
Before solving a problem, decide how the quantities are connected. For a direct relation the proportion has the form a : b = c : x; for an inverse relation the product is constant: a · b = c · x. For example if 5 m of cloth cost 60,000 soms, 8 m cost 8 · 60,000 : 5 = 96,000 soms. If a car at 60 km/h covers a road in 4 hours, at 80 km/h it takes 60 · 4 : 80 = 3 hours. To divide a number in the ratio m : n, split it into (m + n) equal parts and take m parts and n parts.
Worked examples
Divide 120 in the ratio 3 : 5: 3 + 5 = 8 parts, 120 : 8 = 15. The shares are 3 · 15 = 45 and 5 · 15 = 75. Check: 45 + 75 = 120.
A job takes 8 workers 15 days. How long do 12 workers need? Inverse relation: 8 · 15 : 12 = 10 days.
Class activity
“Shares”: groups must share 40 apples between two children in the ratio 3 : 5. They make 8 equal bowls (parts), show how many bowls each child gets and count the apples.
Practice
1
7 kg of sugar cost 70,000 soms. How many soms do 12 kg cost?
120000
2
6 painters paint a fence in 8 days. In how many days will 4 painters do it?
12
3
Divide 120 in the ratio 3 : 5 and find the smaller share.
45
4
With painters reduced from 6 to 4 the days became 12. Does this answer make sense?
Yes: fewer workers need more time (12 > 8). A sense check helps spot mistakes quickly.