24
Directly and inversely proportional quantities
Textbook, pp. 101–105
GoalTell directly proportional and inversely proportional quantities apart.
New words
directly proportional · to‘g‘ri proporsionalinversely proportional · teskari proporsionalcoefficient of proportionality · proporsionallik koeffitsiyentiquantity · miqdor
Explanation
If when one quantity grows several times the other grows the same number of times, they are directly proportional: y = kx, where k is the coefficient of proportionality. An example is price and amount of goods. If when one quantity grows several times the other shrinks the same number of times, they are inversely proportional: x · y = a constant. Examples are the number of workers and the time to finish a job, or speed and travel time. Not every relation is proportional: age and height are not directly proportional.
Worked examples
One notebook costs 3,000 soms: n notebooks cost 3,000 · n. Twice as many notebooks cost twice as much: direct proportion.
6 workers finish a job in 10 days. With twice as many workers it takes 5 days: inverse proportion. The product is constant: 6 · 10 = 12 · 5 = 60.
Class activity
“Which relation?”: the teacher describes a situation (for example “if speed increases, the travel time ...”). Pupils show “direct” (one hand up) or “inverse” (two hands) and give a reason.
Practice
1
How are the number of workers and the time to do a job proportional?
Inversely proportional
2
How are the number of notebooks and their total price proportional?
Directly proportional
3
6 workers finish a job in 10 days. In how many days will 10 workers finish it?
6
4
Are a person's age and height directly proportional? Why?
No. A 10-year-old is not twice as tall as a 5-year-old.