Simplify each of the following expressions:(a) p + p + p + p, p + p + p + q, p + q + p – q(b) p – q + p – q, p + q – p + q(c) p + q – (p + q), p – q – p – q(d) 2d – d – d – d, 2d – d – d – c(e) 2d – d – (d – c), 2d – (d – d) – c(f) 2d – d – c – cSolution:(a) p + p + p + p = 4pp + p + p + q = 3p + qp + q + p – q = 2p

Question

Accepted Answer

Take a look at all the corrected simplest forms (i.e., brackets are removed, like terms are added, and terms with only numbers are also added). Is there any relation between the number of terms and the number of letter-numbers these expressions have?
Solution:
Yes 
If the expression contains a term having only a number,
the number of terms = number of letter-numbers + 1
If an expression has no term that has only numbers,
then number of terms = number of letter-numbers
 4.5 Pick Patterns and Reveal Relationships Formula Detective NCERT In-Text Questions (Pages 95-96) Find out the formula of this number machine.
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 95 Q1]
The formula for the number machine above is “two times the first number minus the second number”. When written as an algebraic expression, the formula is 2a-b. The expression for the first set of inputs is 2 × 5 – 2 = 8. Check that the formula holds for each set of inputs.
Solution:
Yes, the formula holds for each set of inputs.
As, 2 × 8 – 1 = 15; 2 × 9 – 11 = 7; 2 × 10 – 10 = 10; and 2 × 6 – 4 = 8 Find the formulas of the number machines below and write the expression for each set of inputs.
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 95 Q2]
Solution:
The formula for the number machines in the first row is “sum of first number and second number minus two,” and the expression is a + b – 2.
The expression for each set of inputs is:
5 + 2 – 2 = 5, 8 + 1 – 2 = 7, 9 + 11 – 2 = 18, 10 + 10 – 2 = 18, and a + b – 2
The formula for the number machines in the second row is “product of first number and second number plus one,” and the expression is a × b + 1.
The expression for each set of inputs is:
4 × 1 + 1 = 5, 6 × 0 + 1 = 1, 3 × 2 + 1 = 7, 10 × 3 + 1 = 31, and a × b + 1 = ab + 1. Now, make a formula on your own. Write a few number machines as examples using that formula. Challenge your classmates to figure it out!
Solution:
Do it yourself. Algebraic Expressions to Describe Patterns NCERT In-Text Questions (Pages 96-97) Example 12.
Somjit noticed a repeating pattern along the border of a saree.
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 96 Q1]
Use this to find what design appears at positions 99, 122, and 148.
Solution: 
For 99, the remainder on division by 3 is 0, i.e., it is a multiple of 3. So, at position 99, design C will appear.
For 122, the remainder on division by 3 is 2, i.e., it is 1 less than a multiple of 3, i.e., 3n – 1. So, at position 122, design B will appear.
For 148, the remainder on division by 3 is 1, i.e., it is 2 less than a multiple of 3, i.e., 3n – 2. So, at position 148, design A will appear.
 Patterns in a Calendar NCERT In-Text Questions (Page 99) Verify this expression for diagonal sums by considering any 2 × 2 square and taking its top left number to be ‘a’.
Solution:
Let a = 2, then
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 99 Q1]
Here, the diagonal sums are 2 + 10 = 12 and 9 + 3 = 12
And 2a + 8 = 2 × 2 + 8 = 12
Hence, the diagonal sum is equal to 2a + 8. Find the sum of all the numbers. Compare it with the number in the centre: 15. Repeat this for another set of numbers that form this shape. What do you observe?
Solution:
Sum of all numbers = 8 + 14 + 15 + 16 + 22 = 75
The sum is 5 times the number in the centre.
Now, let the number at the centre: 20, then the shape is
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 99 Q2]
Sum of all the numbers = 13 + 19 + 20 + 21 + 27 = 100 = 20 × 5
Further, let the number at the centre: 12, then the shape is
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 99 Q2.1]
Sum of all numbers = 5 + 11 + 12 + 13 + 19 = 60 = 12 × 5
Hence, we see that the total sum is always 5 times the number in the centre. Will this always happen? How do you show this?
Solution:
The general formula for a 3 × 3 square with centre number a is shown here,
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 99 Q3]
Here, the sum is a – 7 + a – 1 + a + a + 1 + a + 7 = a + a + a + a + a – 7 – 1 + 1 + 7
So, the sum of numbers = 5a (5 times the number in the centre). Find other shapes for which the sum of the numbers within the figure is always a multiple of one of the numbers.
Solution:
Do it yourself. Matchstick Patterns Look at the picture below. It is a pattern using matchsticks. Can you identify what the pattern is?
[Expressions using Letter Numbers Class 7 Solutions Ganita Prakash Maths Chapter 4 Page 101 Q1]
We can see that Step 1 has 1 triangle, Step 2 has 2 triangles, Step 3 has 3 triangles, and so on. NCERT In-Text Questions (Page 101) What are these numbers in Step 3 and Step 4?
Solution:
In step 3, there are 3 matchsticks placed horizontally and 4 matchsticks placed diagonally.
In step 4, there are 4 matchsticks placed horizontally and 5 matchsticks placed diagonally. How does the number of matchsticks change in each orientation as the steps increase? Write an expression for the number of matchsticks at Step ‘y in each orientation. Do the two expressions add up to 2y + 1?
Solution:
The pattern in the horizontally placed matchsticks is: 1, 2, 3, 4,….
In general, n for the nth step.
The pattern in the diagonally placed matchsticks is: 2, 3, 4, 5,….
In general, n + 1 for the nth step.
∴ The number of matchsticks placed horizontally at step ‘y’ is y.
The number of matchsticks placed diagonally at step ‘y’ is y + 1.
Now, these two expressions add up to y + y + 1 = 2y + 1. Figure it Out (Pages 102-105) For the problems asking you to find suitable expression(s), first try to understand the relationship between the different quantities in the situation described. If required, assume some values for the unknowns and try to find the relationship.