Class 7 Mathematics · Part 1 · Chapter 2

2. Arithmetic Expressions

Answers to all Figure it Out and in-text questions on terms, brackets, comparing expressions and the distributive property. CBSE / NCERT Class 7 Ganita Prakash.

2.1 Simple Expressions

?Choose your favourite number and write as many expressions as you can having that value.

Let us choose 24.

20 + 4, 30 − 6, 4 × 6, 3 × 8, 48 ÷ 2, 72 ÷ 3, 12 + 12, 50 − 26

FIOFigure it Out

1. Fill in the blanks to make the expressions equal on both sides.

(a) 13 + 4 = 11 + 6 (both are 17)

(b) 22 + 8 = 6 × 5 (both are 30)

(d) 34 − 9 = 25

2. Arrange the expressions in ascending order of their values.

Values: (a) 67 − 19 = 48, (b) 67 − 20 = 47, (c) 35 + 25 = 60, (d) 5 × 11 = 55, (e) 120 ÷ 3 = 40.

Ascending order: 120 ÷ 3 < 67 − 20 < 67 − 19 < 5 × 11 < 35 + 25

?Use ‘>’, ‘<’ or ‘=’ to compare without complicated calculations. Explain your thinking.

(a) 245 + 289 > 246 + 285

245 is 1 less than 246, but 289 is 4 more than 285. So, the left side is 3 more.

(b) 273 − 145 = 272 − 144

The left side starts with 1 more and also takes away 1 more. So, both are equal.

364 is 1 more than 363, but 587 is 2 less than 589. So, the left side is 1 less.

(d) 124 + 245 < 129 + 245

245 is added to both. 124 is less than 129.

(e) 213 − 77 < 214 − 76

The right side starts with 1 more and takes away 1 less. So, it is 2 more.

2.2 Reading and Evaluating Complex Expressions

?Check if replacing subtraction by addition of the inverse does not change the value.

18 − 10 = 8 and 18 + (−10) = 8.

5 − 9 = −4 and 5 + (−9) = −4. The value does not change.

?Why is subtracting a number the same as adding its inverse? (Token Model)

Take 7 − 3. Start with 7 positive tokens and remove 3 of them. 4 tokens are left.

Now add 3 negative tokens to 7 positive tokens. 3 pairs of + and − cancel each other (each pair is 0). Again 4 positive tokens are left.

So, 7 − 3 = 7 + (−3).

?Complete the table.

Expression	As the sum of its terms	Terms
13 − 2 + 6	13 + (−2) + 6	13, −2, 6
5 + 6 × 3	5 + 6 × 3	5, 6 × 3
4 + 15 − 9	4 + 15 + (−9)	4, 15, −9
23 − 2 × 4 + 16	23 + (−2 × 4) + 16	23, −2 × 4, 16
28 + 19 − 8	28 + 19 + (−8)	28, 19, −8

?Does swapping or grouping terms with negative numbers change the sum? Explain with the Token Model.

No. Example: 6 + (−4) = 2 and (−4) + 6 = 2.

Grouping: (−7 + 10) + (−11) = 3 + (−11) = −8 and −7 + (10 + (−11)) = −7 + (−1) = −8.

In the Token Model, we just put all the tokens together. The order or grouping of putting them does not change the final count.

?Does adding the terms of an expression in any order give the same value? Check with more than 3 terms.

Yes. 15 + (−8) + 5 + 12 = 24.

In another order: (15 + 5 + 12) + (−8) = 32 + (−8) = 24.

?Manasa got 11,749 but forgot the fourth number 9055. Does she have to start all over again?

No. The order of adding does not matter. She just adds 9055 to her answer.

11,749 + 9055 = 20,804.

?If the number of friends goes up to 7 and the tip remains the same (dosa ₹23, tip ₹5), how much will they pay? Write the expression and its terms.

Expression: 7 × 23 + 5. Terms: 7 × 23 and 5.

Value = 161 + 5 = ₹166.

?Ruby wrote 6 × 5 + 3 for 33 students in groups of 5. Write expressions for groups of 4 and 7.

Ruby wrote it because 33 students make 6 groups of 5 and 3 students are left out.

Teacher calls ‘4’: 8 × 4 + 1. Terms: 8 × 4, 1.

Teacher calls ‘7’: 4 × 7 + 5. Terms: 4 × 7, 5.

?Identify the terms of the two expressions for ₹432. Think of more ways.

4 × 100 + 1 × 20 + 1 × 10 + 2 × 1 → terms: 4 × 100, 1 × 20, 1 × 10, 2 × 1.

8 × 50 + 1 × 10 + 4 × 5 + 2 × 1 → terms: 8 × 50, 1 × 10, 4 × 5, 2 × 1.

More ways: 4 × 100 + 3 × 10 + 2 × 1; 2 × 100 + 4 × 50 + 6 × 5 + 2 × 1.

FIOFigure it Out

1. Find the values by writing the terms.

(a) 28 − 7 + 8 = 28 + (−7) + 8. Terms: 28, −7, 8. Value = 29

(b) 39 − 2 × 6 + 11 = 39 + (−2 × 6) + 11. Terms: 39, −2 × 6, 11. Value = 39 − 12 + 11 = 38

(d) 48 − 10 × 2 + 16 ÷ 2 = 48 + (−10 × 2) + 16 ÷ 2. Terms: 48, −10 × 2, 16 ÷ 2. Value = 48 − 20 + 8 = 36

(e) 6 × 3 − 4 × 8 × 5 = 6 × 3 + (−4 × 8 × 5). Terms: 6 × 3, −4 × 8 × 5. Value = 18 − 160 = −142

2. Write a story for each expression and find its value.

(a) 89 + 21 − 10: A library had 89 books. It bought 21 more. Then 10 books were lent out. Books left = 100.

(b) 5 × 12 − 6: 5 boxes have 12 eggs each. 6 eggs broke. Eggs left = 60 − 6 = 54.

3. Write the expression, its terms and value.

(a) Elsa has 2 × 100 coins and Anna has 100 ÷ 2 coins. Expression: 2 × 100 + 100 ÷ 2. Terms: 2 × 100, 100 ÷ 2. Value = 200 + 50 = 250 coins.

(b)(i) 4 × 40 + 3 × 20. Terms: 4 × 40, 3 × 20. Value = 160 + 60 = ₹220.

(b)(ii) 2 × (3 × 40). It has one term. Value = 2 × 120 = ₹240.

(c) The window has 2 borders, 6 grills and 7 gaps. Expression: 2 × 3 + 6 × 2 + 7 × 5. Terms: 2 × 3, 6 × 2, 7 × 5. Total height = 6 + 12 + 35 = 53 cm.

Removing Brackets — I

?Tinker the Terms I: fill the blanks with as little computation as possible.

52 + (−16) = 36 (52 is 1 less than 53, so the value is 1 less than 37).

53 + (−15) = 38 (−15 is one more than −16).

53 + (−17) = 36 (−17 is one less than −16).

−87 + (−16) = −103; −88 + (−15) = −103; −86 + (−18) = −104; −97 + (−26) = −123.

FIOFigure it Out

1. Fill in the blanks with numbers and the boxes with operation signs.

(a) 24 + (6 − 4) = 24 + 6 − 4

(b) 38 + (9 − 4) = 38 + 9 − 4

(d) 24 − 6 − 4 = 24 − 6 − 4

(e) 27 − (8 + 3) = 27 − 8 − 3

(f) 27 − (8 − 3) = 27 − 8 + 3

2. Remove the brackets and write the expression having the same value.

(a) 14 + 12 + 10

(b) 14 − 12 − 10

(d) 14 − 12 + 10

(e) −14 + 12 − 10

(f) 14 + 12 + 10

3. Find the values. When are the two expressions equal?

(a) (6 + 10) − 2 = 14 and 6 + (10 − 2) = 14. Equal.

(b) 16 − (8 − 3) = 16 − 8 + 3 = 11 and (16 − 8) − 3 = 5. Not equal.

The two expressions are equal when they have the same terms after removing the brackets.

4. Identify the expressions that have the same value.

(a) 319 − 537 and −537 + 319 (same terms 319 and −537).

(b) The three expressions 87 + 46 − 109 have the same value. Also, 87 − 46 + 109 and (87 − 46) + 109 have the same value.

5. Add brackets to get the values indicated.

(a) 34 − (9 + 12) = 34 − 21 = 13

(b) (56 − 14) − 8 = 42 − 8 = 34

6. Fill the blanks using reasoning.

(a) 423 + 6 = 419 + 10 (423 is 4 more than 419, so the second blank must be 4 more than the first).

(b) 207 − 68 = 210 − 71 (210 is 3 more than 207, so we must take away 3 more).

7. Using 2, 3 and 5 with ‘+’, ‘−’ and brackets, make different values.

2 + 3 + 5 = 10; 3 + 5 − 2 = 6; 2 − 3 + 5 = 4; 2 + 3 − 5 = 0;

3 − 2 − 5 = −4; 2 − (3 + 5) = −6; 5 − (2 + 3) = 0; 3 − (5 − 2) = 0.

So, the values 10, 6, 4, 0, −4 and −6 are possible.

8. Jasoda subtracts 10 and adds 1 to subtract 9.

(a) Yes, she always gets the correct answer. 36 − 9 = 36 − (10 − 1) = 36 − 10 + 1.

(b) To subtract 8, subtract 10 and add 2: 55 − 8 = 55 − 10 + 2 = 47. To subtract 99, subtract 100 and add 1. To add 9, add 10 and subtract 1.

9. Which expressions are equal to (a) 73 − 14 + 1 and (b) 73 − 14 − 1?

73 − 14 + 1 is equal to 73 − (14 − 1) and 73 + (−14 + 1).

73 − 14 − 1 is equal to 73 − (14 + 1) and 73 + (−14 − 1).

Removing Brackets — II

?If Sangmu joins Lhamo and Norbu and orders the same items, what is the expression for the total amount?

3 × (43 + 24) = 3 × 67 = ₹201. It is the same as 3 × 43 + 3 × 24.

?5 × 4 + 3 ≠ 5 × (4 + 3). Can you explain why? Is 5 × (4 + 3) = 5 × (3 + 4) = (3 + 4) × 5?

5 × 4 + 3 = 20 + 3 = 23, but 5 × (4 + 3) = 5 × 7 = 35. In the first, only 4 is multiplied by 5.

Yes, all three are equal to 35. Swapping terms or swapping numbers in a product does not change the value.

?Find 97 × 25. Use this method to find 95 × 8, 104 × 15 and 49 × 50.

97 × 25 = 100 × 25 − 3 × 25 = 2500 − 75 = 2425

(a) 95 × 8 = 100 × 8 − 5 × 8 = 800 − 40 = 760

(b) 104 × 15 = 100 × 15 + 4 × 15 = 1500 + 60 = 1560

Yes, it is quicker. It works well when one number is close to 10, 50, 100, 1000, etc. Example: 98 × 7 = 700 − 14 = 686.

FIOFigure it Out

1. Fill in the blanks and boxes.

(a) 3 × (6 + 7) = 3 × 6 + 3 × 7

(b) (8 + 3) × 4 = 8 × 4 + 3 × 4

(d) (9 + 2) × 4 = 9 × 4 + 2 × 4

(e) 3 × (10 + 4) = 3 × 10 + 3 × 4 (any number can be used in place of 10)

(f) (13 + 6) × 4 = 13 × 4 + 6 × 4

(g) 3 × (5 + 2) = 3 × 5 + 3 × 2

(h) (2 + 3) × 4 = 2 × 4 + 3 × 4

(i) 5 × (9 − 2) = 5 × 9 − 5 × 2

(j) (5 − 2) × 7 = 5 × 7 − 2 × 7

(k) 5 × (8 − 3) = 5 × 8 − 5 × 3

(l) (8 − 3) × 7 = 8 × 7 − 3 × 7

(m) 5 × (12 − 3) = 5 × 12 − 5 × 3 (any number can be used in place of 3)

(n) (15 − 6) × 7 = 15 × 7 − 6 × 7

(o) 5 × (9 − 4) = 5 × 9 − 5 × 4

(p) (17 − 9) × 7 = 17 × 7 − 9 × 7

2. Fill ‘<’, ‘>’ or ‘=’ by reasoning.

(a) (8 − 3) × 29 > (3 − 8) × 29, because 8 − 3 is positive and 3 − 8 is negative.

(b) 15 + 9 × 18 < (15 + 9) × 18, because on the right, 15 is also multiplied by 18.

(d) (34 − 28) × 42 = 34 × 42 − 28 × 42 (multiple of a difference).

3. Other ways of getting 14.

(a) 2 × (3 + 4) = 14

(b) 2 × (2 + 5) = 14

(d) 1 × (10 + 4) = 14

4. Find the sum of the numbers in each picture in two different ways.

Picture 1 (five 4s and four 8s): 5 × 4 + 4 × 8 = 20 + 32 = 52. Another way, row by row: (4 + 8 + 4) + (8 + 4 + 8) + (4 + 8 + 4) = 16 + 20 + 16 = 52.

Picture 2 (eight 5s and eight 6s): 8 × 5 + 8 × 6 = 40 + 48 = 88. Another way: 8 × (5 + 6) = 8 × 11 = 88.

FIOFigure it Out

1. Write expressions and find their values.

(a) 7 × (9 + 11) = 7 × 20 = 140 kg

(b) 12 × (20,000 − (5,000 + 5,000 + 2,000)) = 12 × 8,000 = ₹96,000

(c) In one day and night, the snail climbs 3 − 2 = 1 cm. After 7 days and nights, it is at 7 cm. On the 8th day it climbs 3 cm and reaches 10 cm: 7 × (3 − 2) + 3 = 10. So, it gets the treat on the 8th day.

2. Melvin reads one story every day except Tuesdays and Saturdays. How many stories in 8 weeks? Which expressions describe this?

He reads 5 stories a week. In 8 weeks: 5 × 8 = 40 stories.

Expressions (b) (7 − 2) × 8 and (g) 7 × 8 − 2 × 8 describe this.

3. Find different ways of evaluating.

(a) Pairing: (1 − 2) + (3 − 4) + (5 − 6) + (7 − 8) + (9 − 10) = 5 × (−1) = −5. Or, (1 + 3 + 5 + 7 + 9) − (2 + 4 + 6 + 8 + 10) = 25 − 30 = −5.

(b) Pairing: (1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) = 0. Or, (1 + 1 + 1 + 1 + 1) − (1 + 1 + 1 + 1 + 1) = 5 − 5 = 0.

4. Compare using ‘<’, ‘>’ or ‘=’.

(a) 49 − 7 + 8 = 49 − 7 + 8 (the same expression)

(b) 83 × 42 − 18 > 83 × 40 − 18 (83 × 42 is more than 83 × 40)

(d) 23 × 48 − 35 > 23 × (48 − 35) (the right side subtracts 23 × 35)

(e) (16 − 11) × 12 = −11 × 12 + 16 × 12

(f) (76 − 53) × 88 > 88 × (53 − 76) (the right side is negative)

(g) 25 × (42 + 16) = 25 × (43 + 15) (42 + 16 = 43 + 15 = 58)

(h) 36 × (28 − 16) > 35 × (27 − 15) (both brackets are 12, and 36 > 35)

5. Identify the expressions equal to the given expression.

(a) 83 − 37 − 12 is equal to (i) 84 − 38 − 12 and (iv) −37 + 83 − 12.

(b) 93 + 37 × 44 + 76 is equal to (iv) 37 × 44 + 93 + 76 only.

6. Choose a number and create ten different expressions having that value.

Number 26: 10 + 16, 30 − 4, 13 × 2, 52 ÷ 2, 5 × 5 + 1, 3 × 7 + 5, 100 − 74, 2 × (10 + 3), 6 × 4 + 2, 40 − 2 × 7.

Puzzle: Expression Engineer!

?Using four 4’s, create expressions to get the values from 1 to 20.

1 = 44 ÷ 44; 2 = 4 ÷ 4 + 4 ÷ 4; 3 = (4 + 4 + 4) ÷ 4; 4 = 4 + 4 × (4 − 4); 5 = (4 × 4 + 4) ÷ 4;

6 = 4 + (4 + 4) ÷ 4; 7 = 44 ÷ 4 − 4; 8 = 4 + 4 + 4 − 4; 9 = 4 + 4 + 4 ÷ 4; 10 = (44 − 4) ÷ 4;

12 = (44 + 4) ÷ 4; 15 = 44 ÷ 4 + 4; 16 = 4 + 4 + 4 + 4; 17 = 4 × 4 + 4 ÷ 4; 20 = 4 × (4 + 4 ÷ 4).

11, 13, 14, 18 and 19 cannot be made using only +, −, ×, ÷ and brackets. They need other tools, like the square root of 4.

?Using 1, 2, 3, 4 and 5 exactly once, get values between −10 and +10.

Some examples: 1 + 2 + 3 + 4 − 5 = 5; 1 − 2 − 3 − 4 + 5 = −3; (1 + 2) × 3 − 4 − 5 = 0;

5 × 2 − 4 − 3 + 1 = 4; 1 + 2 − 3 − 4 − 5 = −9; 5 + 4 − 3 × 2 − 1 = 2.

?Using the numbers 0 to 9 exactly once, make an expression with a value 100.

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9 = 28 + 72 = 100

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