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.
Answer: a = 60, b = 200, so y = 60x + 200 (₹60 per hour, ₹200 fixed fee).
Step-by-step solution
Given: x = 10, y = 800; x = 15, y = 1100
To find: a and b in y = ax + b
To find: a and b in y = ax + b
Idea: Two observations give two equations in a and b. Substitute b from the first into the second.
- 800 = 10a + b … (1) and 1100 = 15a + b … (2).1 mark
- From (1), b = 800 − 10a. In (2): 1100 = 15a + 800 − 10a ⇒ 5a = 300 ⇒ a = 60.1 mark
- b = 800 − 10 × 60 = 800 − 600 = 200. So y = 60x + 200.1 mark
a = 60 and b = 200 (the bill is y = 60x + 200).
Check: x = 10: 600 + 200 = 800 ✓. x = 15: 900 + 200 = 1100 ✓.
Answer to write in the exam
800 = 10a + b … (1); 1100 = 15a + b … (2)
From (1): b = 800 − 10a
In (2): 1100 = 15a + 800 − 10a ⇒ 5a = 300 ⇒ a = 60
b = 800 − 600 = 200
∴ a = 60, b = 200
Common mistakes that cost marks
- Taking a = 800 ÷ 10 = 80, ignoring the fixed monthly fee.
- Writing 5a = 300 as a = 1500.
- Mixing up a and b in the final answer (saying the fixed fee is ₹60).
How this can come in the exam
MCQ (1 mark)
For the gym above (y = 60x + 200), the bill for 22 hours is
- ₹1320
- ₹1520
- ₹1540
- ₹1720
Show answer
(B) ₹1520
60 × 22 + 200 = 1320 + 200 = ₹1520.
Short answer (3 marks)
A cycle rental shop charges a fixed fee plus an hourly rate. 3 hours cost ₹170 and 7 hours cost ₹330. Find the fee and the rate, and the cost for 10 hours.
Show answer
4a = 160 ⇒ a = ₹40 per hour (1 mark). b = 170 − 120 = ₹50 (1 mark). 10 hours: 400 + 50 = ₹450 (1 mark).Try one yourself
A swimming pool charges a fixed fee plus a cost per visit. 4 visits cost ₹380 and 10 visits cost ₹650. Find a and b in y = ax + b.
Show answer
6a = 270 ⇒ a = 45; b = 380 − 180 = 200.
More questions like this
- Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by °C = a °F + b. Find a and b, given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit.
(Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this information to find a and b, and thus, the linear relationship between °C and °F.) - Identify other points on the line by completing the following table.
x: 1, 2, 5, 7, 9, 12, 20
y: 3, __, __, 15, __, __, __ - Let us plot the points (−1, −3), (0, 0), (1, 3), (3, 9), (4, 12) in the coordinate plane on a graph paper as shown in the figure. Join the points (−1, −3) and (4, 12) using a ruler. Doing so, observe that all five points lie on a straight line. Can you guess the equation of this line by looking at the relationship between the x and y coordinates of each point?
- Let us plot the points (− 3, 6), (− 2, 4), (0, 0), (1, − 2), (2, − 4), (3, − 6) in the coordinate plane on a graph paper as shown in the figure. Join the points (− 3, 6) and (3, − 6) using a ruler. Doing so, observe that all five points lie on a straight line. Can you guess the equation of this line by looking at the relationship between the x and y coordinates of each point?
- Draw the graphs of y = 12x, y = x, y = 2x by selecting suitable points on these lines.
(Hint: In order to graph y = 12x, we could take the points (0, 0) and (4, 2). Can you verify that these lie on the line?)