A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was ₹400. When she accessed 14 modules, her bill was ₹500. If the monthly bill y depends on the number of modules accessed, x, according to the relation y = ax + b, find the values of a and b.
Step-by-step solution
To find: a and b in y = ax + b
Idea: Put both observations into y = ax + b to get two equations, then substitute b from the first into the second.
- 400 = 10a + b … (1) and 500 = 14a + b … (2).1 mark
- From (1), b = 400 − 10a. In (2): 500 = 14a + 400 − 10a ⇒ 4a = 100 ⇒ a = 25.1 mark
- b = 400 − 10 × 25 = 400 − 250 = 150. So y = 25x + 150.1 mark
Check: x = 10: 250 + 150 = 400 ✓. x = 14: 350 + 150 = 500 ✓.
Answer to write in the exam
400 = 10a + b … (1); 500 = 14a + b … (2)
From (1): b = 400 − 10a
In (2): 500 = 14a + 400 − 10a ⇒ 4a = 100 ⇒ a = 25
b = 400 − 250 = 150
∴ a = 25, b = 150
Common mistakes that cost marks
- Taking a = 400 ÷ 10 = 40, ignoring the fixed fee.
- Subtracting the equations wrongly: 4a = 100 but then a = 400.
- Giving b = 400 − 25 = 375 (forgetting to multiply a by 10).
How this can come in the exam
A bill y = ax + b is ₹260 for 8 items and ₹320 for 12 items. The value of b is
- ₹15
- ₹140
- ₹120
- ₹20
Show answer
(B) ₹140
4a = 60 ⇒ a = 15; b = 260 − 120 = 140.
A music app charges a fixed fee plus a cost per song downloaded. 6 songs cost ₹138 and 15 songs cost ₹210. Find the fixed fee and the cost per song.
Show answer
138 = 6a + b, 210 = 15a + b (1 mark). 9a = 72 ⇒ a = ₹8 per song (1 mark). b = 138 − 48 = ₹90 fixed fee (1 mark).Try one yourself
A library charges a fixed fee plus a cost per book. 5 books cost ₹95 and 9 books cost ₹135. Find the fee and the cost per book.
Show answer
4a = 40 ⇒ a = ₹10 per book; b = 95 − 50 = ₹45.
More questions like this
- A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student using the gym observed that when she used the badminton court for 10 hours, her bill was ₹800. When she used it for 15 hours, her bill was ₹1100. If the monthly bill y depends on the hours of the use of the badminton court, x, according to the relation y = ax + b, find the values of a and b.
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x: 1, 2, 5, 7, 9, 12, 20
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