Rakesh’s puzzle: “I have thought of two numbers”, he said. “Their sum is 25 and their difference is 11”.
We can also solve this using a pair of linear equations.
Answer: x + y = 25 and x − y = 11 give x = 18, y = 7. The numbers are 18 and 7.
Step-by-step solution
To find: The two numbers
Idea: Each sentence gives one linear equation. Adding the two equations removes y in one step.
- Let the numbers be x and y (with x the larger). Sum 25: x + y = 25 … (1). Difference 11: x − y = 11 … (2).½ mark
- Add (1) and (2): 2x = 36, so x = 18.½ mark
- From (1): y = 25 − 18 = 7.½ mark
- The numbers are 18 and 7, the same pair found by listing combinations in a table (10, 15 → difference −5; 20, 5 → 15; 18, 7 → 11 ✓).½ mark
The two numbers are 18 and 7.
Check: 18 + 7 = 25 ✓, 18 − 7 = 11 ✓.
Answer to write in the exam
Let the numbers be x, y
x + y = 25 … (1); x − y = 11 … (2)
(1) + (2): 2x = 36 ⇒ x = 18
y = 25 − 18 = 7
∴ The numbers are 18 and 7
Common mistakes that cost marks
- Writing the difference as y − x = 11 while calling x the larger number; this gives y = 18, x = 7, the same numbers with the letters swapped, and causes confusion. Decide which letter is larger first.
- After 2x = 36, forgetting to halve and giving 36 as a number.
- Answering only one of the two numbers.
How this can come in the exam
MCQ (1 mark)
Two numbers have sum 40 and difference 14. The larger number is
- 27
- 26
- 13
- 20
Show answer
(A) 27
2x = 54 ⇒ x = 27.
Try one yourself
The sum of two numbers is 52 and their difference is 18. Find them.
Show answer
2x = 70 ⇒ 35 and 17.
More questions like this
- Consider another situation. A science exhibition charges an entry fee that is different for adults and children. Here are the amount paid by two different groups.
Group A: 2 adult tickets and 3 child tickets for ₹600
Group B: 3 adult tickets and 2 child tickets for ₹700
How much do individual adult and child tickets cost?
If the cost of one adult ticket is ₹x and one child ticket is ₹y, then: 2x + 3y = 600, 3x + 2y = 700.
This gives us another pair of linear equations in two variables. A solution to these equations will give us the answer that we seek. Can you see why? - Given below are some everyday situations. Construct a pair of linear equations in two variables for each situation:
On two different days a family buys movie tickets and snack boxes. Let the cost of one movie ticket be ₹x and the cost of one snack box be ₹y. Frame a pair of linear equations in x and y to represent the following situations: - Two friends, Sahil and Meena, travelled by taxi. Let the fixed charge for one taxi trip be ₹x and the additional charge per kilometre be ₹y. Frame a pair of linear equations in x and y to represent the following situation:
- In a sports meet, tickets for adults cost ₹150 each and tickets for children cost ₹100 each. A total of 200 people attended the meet, and the total amount collected from ticket sales was ₹25,000. (Let the number of adult tickets sold be x and the number of children’s tickets sold be y.) Frame a pair of linear equations in x and y to represent this situation.
- Solve the puzzle and discuss your strategy with your friends. Can you determine the value represented by each shape?
All Linear equations in two variables questions · All maths questions