Raju went to a shop where there were sealed boxes of different colours on sale. The shop owner told him that the red boxes have 4 pens each and the blue boxes have 5 pencils each. Now, if Raju bought x red boxes and y blue boxes, how can he quickly figure out the total quantity of pens and pencils? Also, if he got 3 extra pens free, how many pens and pencils did he get altogether?
Step-by-step solution
To find: An expression for the total number of pens and pencils
Idea: When we do not know a number, we use a letter for it. “4 pens in each of x boxes” means 4 × x = 4x pens. Then we add all the groups.
- Pens: each red box has 4 pens, so x red boxes have 4 × x = 4x pens.½ mark
- Pencils: each blue box has 5 pencils, so y blue boxes have 5 × y = 5y pencils.½ mark
- So the boxes hold 4x + 5y pens and pencils. Raju can find the total quickly by putting his numbers of boxes in place of x and y.
- He also gets 3 free pens, so we add 3:
Total number of pens and pencils = 4x + 5y + 3.1 mark - Words to know: 4x, 5y and 3 are the terms; x and y are the variables; 4 and 5 are the coefficients of x and y; 3 is the constant.
Check: Suppose Raju buys 2 red boxes and 3 blue boxes. Counting directly: 8 pens + 15 pencils + 3 free pens = 26. The expression gives 4(2) + 5(3) + 3 = 8 + 15 + 3 = 26 ✓.
Answer to write in the exam
Pens in x red boxes = 4 × x = 4x
Pencils in y blue boxes = 5 × y = 5y
Free pens = 3
∴ Total number of pens and pencils = 4x + 5y + 3
Common mistakes that cost marks
- Writing x + 4 instead of 4x. There are 4 pens in each of the x boxes, so we multiply: 4 × x.
- Adding the free pens to the coefficient, as in 7x + 5y. The 3 free pens are a fixed number, not 3 more per box, so they are a separate constant term +3.
- Writing 9xy by “combining” 4x and 5y. They are unlike terms (different variables), so they cannot be added into one term.
How this can come in the exam
A packet of biscuits costs ₹30 and a packet of chips costs ₹20. The cost of p packets of biscuits and q packets of chips, with a ₹15 delivery charge, is
- ₹(50pq + 15)
- ₹(30p + 20q + 15)
- ₹(30p + 20q)
- ₹(30 + p + 20 + q + 15)
Show answer
(B) ₹(30p + 20q + 15)
Biscuits cost 30 × p = 30p, chips 20 × q = 20q, and the delivery charge is a constant 15.
In the expression 6a + 9b + 4, write (i) the terms, (ii) the variables, (iii) the coefficients of a and b, (iv) the constant.
Show answer
(i) Terms: 6a, 9b, 4 (½ mark). (ii) Variables: a and b (½ mark). (iii) Coefficient of a is 6, of b is 9 (½ mark). (iv) Constant: 4 (½ mark).Try one yourself
A notebook has 80 pages and a diary has 120 pages. Meena has m notebooks and d diaries, and 25 loose sheets. Write an expression for the total number of pages and sheets.
Show answer
80m + 120d + 25.
More questions like this
- A rectangular garden of length l metres and width w metres has to be fenced and decorated. A wire fence is to be laid along the length costing ₹100 per metre and a wooden fence is to be built along the width costing ₹80 per metre. Special seeds have to be sown throughout the garden which will cost ₹50 per square metre. What will be the total cost incurred?
- Thus, 200l + 160w + 50lw is the algebraic expression for the total cost.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. How is it different from the algebraic expression 4x + 5y + 3 for Raju’s pens and pencils? - A wire of length 20 cm is bent in different ways to form rectangles. For example, we can have a rectangle with length 7 cm and width 3 cm. We can also have one of length 5.5 cm and width 4.5 cm. (Think of a few more ways of forming such rectangles.) Can you write an expression for the area of such rectangles?
- The expression for the area of these rectangles is x(10 − x) or 10x − x2.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. Can you point out any similarity or difference between the algebraic expressions obtained for Raju’s pens and pencils (4x + 5y + 3) and for these rectangles? - Find the degrees of the following polynomials: