A telecom company charges a fixed monthly fee and an additional cost per GB of the internet data used. A student observes that when she used 10 GB, her bill was ₹350. When she used 20 GB, her bill was ₹550. If the monthly bill y depends on the amount of data used, x (in GB), according to the relation y = ax + b, find the values of a and b.
Step-by-step solution
To find: a and b
Idea: Each observation gives one equation in a and b. Make b the subject of the first and substitute it into the second (substitution).
- Substitute the two observations into y = ax + b: 350 = 10a + b and 550 = 20a + b.1 mark
- From the first: b = 350 − 10a. Substitute into the second: 550 = 20a + (350 − 10a).½ mark
- 550 = 10a + 350 ⇒ 10a = 200 ⇒ a = 20.½ mark
- b = 350 − 10 × 20 = 350 − 200 = 150.½ mark
- So y = 20x + 150 is the linear relationship between the bill y (₹) and the data x (GB).½ mark
Check: x = 10: 200 + 150 = 350 ✓. x = 20: 400 + 150 = 550 ✓.
Answer to write in the exam
y = ax + b
350 = 10a + b … (1); 550 = 20a + b … (2)
From (1): b = 350 − 10a
In (2): 550 = 20a + 350 − 10a ⇒ 10a = 200 ⇒ a = 20
b = 350 − 200 = 150
∴ a = 20, b = 150; y = 20x + 150
Common mistakes that cost marks
- Dividing 350 by 10 to get a = 35. That ignores the fixed fee b.
- Substituting x and y the wrong way round, writing 10 = 350a + b.
- Finding a and stopping without finding b.
How this can come in the exam
If y = ax + b, y = 9 when x = 2 and y = 21 when x = 6, then a is
- 3
- 4.5
- 12
- 2
Show answer
(A) 3
Subtract: 4a = 12, so a = 3 (and b = 3).
An electricity board charges a fixed monthly fee plus a rate per unit. A family used 100 units and paid ₹750; next month they used 160 units and paid ₹1110. The bill is y = ax + b for x units.
(i) Write two equations. (ii) Find a. (iii) Find b. (iv) Find the bill for 200 units.
Show answer
(i) 750 = 100a + b, 1110 = 160a + b (1 mark). (ii) 60a = 360 ⇒ a = 6 (1 mark). (iii) b = 750 − 600 = 150 (1 mark). (iv) 6 × 200 + 150 = ₹1350 (1 mark).Try one yourself
A plumber charges a call-out fee plus a rate per hour. A 2-hour job costs ₹700 and a 5-hour job costs ₹1450. Find the rate and the fee.
Show answer
3a = 750 ⇒ a = ₹250 per hour; b = 700 − 500 = ₹200.
More questions like this
- Thus, y = 20x + 150 represents the linear relationship between y, the bill amount in Rupees, and x, the number of GB of the internet data used.
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