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Class 7 Ganita Prakash · Chapter 2
Class 7 Ganita Prakash · Chapter 2
Arithmetic Expressions
Page-wise textbook solutions (pages 24–45) · Every step · Figures · Common mistakes · Tips
✔ Understood 0/105
๐️ Key ideas of this chapter
- An expression like 30 + 5 × 4 has a value (here 50).
- Terms are the parts joined by + signs. Change subtraction to adding the inverse: 83 − 14 = 83 + (−14). A product like 5 × 4 is one term.
- Find each term first, then add the terms. Brackets are worked out first.
- Swapping terms (commutative) or grouping terms (associative) does not change the sum.
- Minus before a bracket changes every sign inside: 100 − (15 + 56) = 100 − 15 − 56. Plus before a bracket changes nothing.
- Distributive property: a × (b + c) = a × b + a × c; a × (b − c) = a × b − a × c.
๐ Textbook Page 24
๐ Page 24Think
Choose your favourite number and write as many expressions as you can having that value.
1Example: favourite number 24.
2Using +: 20 + 4, 12 + 12, 15 + 9
3Using −: 30 − 6, 50 − 26
4Using ×: 4 × 6, 3 × 8, 2 × 12
5Using ÷: 48 ÷ 2, 72 ÷ 3, 96 ÷ 4
✅ Answer: For 24: 20 + 4, 30 − 6, 4 × 6, 48 ÷ 2, 3 × 8 … (your own number gives your own list).
๐ Textbook Page 25
๐ Page 25Figure it Out · Q1(a)
Fill in the blank to make both sides equal: 13 + 4 = ____ + 6
113 + 4 = 17
217 − 6 = 11, so 11 + 6 = 17 ✔
✅ Answer: 11
๐ Page 25Figure it Out · Q1(b)
Fill in the blank to make both sides equal: 22 + ____ = 6 × 5
16 × 5 = 30
230 − 22 = 8, so 22 + 8 = 30 ✔
✅ Answer: 8
๐ Page 25Figure it Out · Q1(c)
Fill in the blank to make both sides equal: 8 × ____ = 64 ÷ 2
164 ÷ 2 = 32
232 ÷ 8 = 4, so 8 × 4 = 32 ✔
✅ Answer: 4
๐ Page 25Figure it Out · Q1(d)
Fill in the blank to make both sides equal: 34 − ____ = 25
134 − 25 = 9, so 34 − 9 = 25 ✔
✅ Answer: 9
๐ Page 25Figure it Out · Q2
Arrange in ascending order of value: (a) 67 − 19 (b) 67 − 20 (c) 35 + 25 (d) 5 × 11 (e) 120 ÷ 3
1Find each value: 67 − 19 = 48, 67 − 20 = 47, 35 + 25 = 60, 5 × 11 = 55, 120 ÷ 3 = 40
2Order the values: 40 < 47 < 48 < 55 < 60
✅ Answer: 120 ÷ 3 < 67 − 20 < 67 − 19 < 5 × 11 < 35 + 25
๐ก Tip: 67 − 20 is less than 67 − 19: taking away MORE leaves LESS.
๐ Page 25Example 2
Which is greater: 1023 + 125 or 1022 + 128? (Try without adding.)
1Think of Raja with 1023 marbles and Joy with 1022: Raja has 1 more.
2Raja gets 125 more; Joy gets 128 more: Joy gets 3 more.
3So in the end Joy has 3 − 1 = 2 more.
✅ Answer: 1023 + 125 < 1022 + 128 (the right side is 2 more).
๐ Textbook Page 26
๐ Page 26Compare (a)
Use >, < or = without long calculation: 245 + 289 ___ 246 + 285
1246 is 1 more than 245, but 285 is 4 less than 289 → left side is 3 more.
2Check: 245 + 289 = 534 and 246 + 285 = 531.
✅ Answer: 245 + 289 > 246 + 285
๐ Page 26Compare (b)
Use >, < or = without long calculation: 273 − 145 ___ 272 − 144
1Start with 1 less AND take away 1 less → the answers are equal.
2Check: 273 − 145 = 128 and 272 − 144 = 128.
✅ Answer: 273 − 145 = 272 − 144
๐ Page 26Compare (c)
Use >, < or = without long calculation: 364 + 587 ___ 363 + 589
1363 is 1 less, 589 is 2 more → right side is 1 more.
2Check: 364 + 587 = 951 and 363 + 589 = 952.
✅ Answer: 364 + 587 < 363 + 589
๐ Page 26Compare (d)
Use >, < or = without long calculation: 124 + 245 ___ 129 + 245
1Same 245 is added; 129 > 124 → right side is 5 more.
2Check: 124 + 245 = 369 and 129 + 245 = 374.
✅ Answer: 124 + 245 < 129 + 245
๐ Page 26Compare (e)
Use >, < or = without long calculation: 213 − 77 ___ 214 − 76
1Right side starts with 1 more and takes away 1 less → right side is 2 more.
2Check: 213 − 77 = 136 and 214 − 76 = 138.
✅ Answer: 213 − 77 < 214 − 76
๐ Textbook Page 27
๐ Page 27Example 4
Mallesh brought 30 marbles. Arun brought 5 bags with 4 marbles in each. Purna says 30 + 5 × 4 = 140 and Mallesh says it is 50. Who is right?
1Arun’s marbles = 5 × 4 = 20.
2Total = 30 + 20 = 50.
3Purna added 30 + 5 first — but 5 is the number of BAGS, not marbles.
✅ Answer: Mallesh is right: 30 + (5 × 4) = 50.
⚠️ Common mistake: Working from left to right without thinking: (30 + 5) × 4 = 140 is wrong here.
๐ Page 27Example 5
Irfan buys biscuits for ₹15 and toor dal for ₹56. He gives ₹100. Write an expression for the change and find it.
1Total cost = 15 + 56.
2Change = 100 − (15 + 56).
3100 − 71 = 29.
✅ Answer: 100 − (15 + 56) = ₹29.
⚠️ Common mistake: Writing 100 − 15 + 56 = 141 — he cannot get back more than he paid! Use brackets.
๐ Textbook Page 28
๐ Page 28Check
Check, with different examples, that subtracting a number is the same as adding its inverse.
118 − 10 = 8 and 18 + (−10) = 8 ✔
25 − 9 = −4 and 5 + (−9) = −4 ✔
3−3 − 4 = −7 and −3 + (−4) = −7 ✔
✅ Answer: Yes — in every case a − b = a + (−b).
๐ Page 28Try This · Token model
Use the token model of integers to explain why subtracting a number is the same as adding its inverse.
1Take 5 positive tokens (+). Subtracting 3 means removing 3 positive tokens → 2 positive tokens are left.
2Now instead ADD 3 negative tokens (−3) to the 5 positive tokens.
3Each negative token cancels one positive token (a zero pair). 3 zero pairs vanish → 2 positive tokens are left.
4Both ways leave the same result, so 5 − 3 = 5 + (−3).
✅ Answer: Removing a token and adding its opposite token have the same effect, so a − b = a + (−b).
๐ Textbook Page 29
๐ Page 29Complete the table
Write each expression as a sum of its terms and list the terms: 5 + 6 × 3, 4 + 15 − 9, 23 − 2 × 4 + 16, 28 + 19 − 8.
1Change every subtraction into adding the inverse.
2Terms are the parts separated by + signs. A product like 6 × 3 is ONE term.
| Expression | As a sum of 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 |
✅ Answer: See the completed table.
๐ Page 29Example 6
A drone goes 6 m up and then 4 m down. Write the expression as a sum of terms. Does swapping the terms change the value?
16 − 4 = 6 + (−4) = 2
2Swap: (−4) + 6 = 2
3Same value.
6 + −4 = 2
−4 + 6 = 2
✅ Answer: 6 + (−4) = (−4) + 6 = 2 m above the terrace.
๐ Page 29Check
Does swapping two terms change the sum when some terms are negative? Check with more examples.
1(−4) + (−2) = −6 and (−2) + (−4) = −6
2(−9) + 5 = −4 and 5 + (−9) = −4
312 + (−7) = 5 and (−7) + 12 = 5
✅ Answer: No. Swapping two terms never changes the sum (commutative property of addition).
๐ Textbook Page 30
๐ Page 30Try This · Token model
Use tokens to explain why swapping terms does not change the sum.
1(−4) + 6: put 4 negative tokens, then 6 positive tokens. 4 zero pairs vanish → 2 positive tokens.
26 + (−4): put 6 positive tokens first, then 4 negative tokens. The same 4 zero pairs vanish → 2 positive tokens.
3The pile has the same tokens either way; only the order of putting them in changed.
✅ Answer: The final heap is the same, so the sum is the same.
๐ Page 30Grouping
Add (−7) + 10 + (−11) in two ways: (first two, then the third) and (last two, then the first).
1Way 1: (−7) + 10 = 3; then 3 + (−11) = −8.
2Way 2: 10 + (−11) = −1; then (−7) + (−1) = −8.
3Way 3: (−7) + (−11) = −18; then −18 + 10 = −8.
✅ Answer: All ways give −8 (associative property of addition).
๐ Page 30Check
Check grouping with negative numbers using another example.
1(−6) + (−7) + (−13):
2[(−6) + (−7)] + (−13) = −13 + (−13) = −26
3(−6) + [(−7) + (−13)] = −6 + (−20) = −26
✅ Answer: Yes, grouping does not change the sum.
๐ Textbook Page 31
๐ Page 31Check
Does adding the terms in any order give the same value? Try an expression with 4 terms.
115 + (−8) + 5 + 12
2Order 1: 15 + (−8) = 7; 7 + 5 = 12; 12 + 12 = 24
3Order 2: 15 + 5 + 12 = 32; 32 + (−8) = 24
4Order 3: (−8) + 12 = 4; 4 + 15 + 5 = 24
✅ Answer: Yes — any order gives 24.
๐ Page 31Evaluate
Find the value of 5 × (3 + 2) + 7 × 8 + 3.
1Terms: 5 × (3 + 2), 7 × 8, 3
25 × (3 + 2) = 5 × 5 = 25
37 × 8 = 56
425 + 56 + 3 = 84
5 × (3 + 2) + 7 × 8 + 3 = 84
✅ Answer: 84
๐ Page 31Manasa’s sum
Manasa added a long list and got 11749, but forgot the fourth number 9055. Does she have to start again?
1The order of adding does not matter (commutative and associative properties).
2So she can just add the missing number to her total.
311749 + 9055 = 20804
✅ Answer: No. The correct total is 11749 + 9055 = 20804.
๐ Textbook Page 32
๐ Page 32Example 7
4 friends order dosas at ₹23 each and give a ₹5 tip: 4 × 23 + 5. If 7 friends come and the tip is the same, write the expression and its terms.
1Cost of 7 dosas = 7 × 23 = 161
2Expression: 7 × 23 + 5
3Terms: 7 × 23 and 5
4Value: 161 + 5 = 166
✅ Answer: 7 × 23 + 5 = ₹166; terms 7 × 23 and 5.
๐ Page 32Example 8
33 students play “Fire in the mountain”. The teacher calls ‘5’ and Ruby writes 6 × 5 + 3. Why?
133 students make groups of 5.
26 groups use 6 × 5 = 30 students.
33 students are left out.
4So 33 = 6 × 5 + 3 (3 more than 6 × 5).
✅ Answer: Because 6 groups of 5 are formed and 3 students are out.
๐ Textbook Page 33
๐ Page 33Example 8
What would Ruby write if the teacher called out ‘4’? And ‘7’? Identify the terms.
133 ÷ 4: 8 groups of 4 = 32, 1 left → 8 × 4 + 1 (terms 8 × 4 and 1)
233 ÷ 7: 4 groups of 7 = 28, 5 left → 4 × 7 + 5 (terms 4 × 7 and 5)
✅ Answer: ‘4’ → 8 × 4 + 1; ‘7’ → 4 × 7 + 5
๐ก Tip: For your own class of n students: groups × size + students left over.
๐ Page 33Example 9
Raghu packs 100 kg of rice into 2 kg packets. He already had four 2 kg packets. Write an expression for the number of packets and identify the terms.
1New packets = 100 ÷ 2 = 50
2Total = 4 + 100 ÷ 2
3Terms: 4 and 100 ÷ 2
4Value = 4 + 50 = 54
✅ Answer: 4 + 100 ÷ 2 = 54 packets.
๐ Page 33Example 10
₹432 = 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1 and 432 = 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1. Identify the terms. Find more ways.
1Way 1 terms: 4 × 100, 1 × 20, 1 × 10, 2 × 1
2Way 2 terms: 8 × 50, 1 × 10, 4 × 5, 2 × 1
3More ways: 40 × 10 + 3 × 10 + 2 × 1; 4 × 100 + 6 × 5 + 2 × 1
✅ Answer: Terms are the products joined by + signs. Another way: 4 × 100 + 6 × 5 + 2 × 1 = 432.
๐ Textbook Page 34
๐ Page 34Example 11
Which arrangement matches 5 × 2 + 3? Write an expression for the other arrangement (2 rows, each with 5 yellow and 3 blue squares).
15 × 2 + 3 = 10 + 3 = 13 → two rows of 5 and 3 extra.
2The other picture has 2 rows of (5 + 3) squares: 2 × (5 + 3) = 16.
3It can also be written 5 + 3 + 5 + 3 or 5 × 2 + 3 × 2.
✅ Answer: 5 × 2 + 3 matches the left picture; the right picture is 2 × (5 + 3) = 16.
๐ Page 34Figure it Out · Q1(a)
Find the value of 28 − 7 + 8 by writing its terms.
1Terms: 28, −7, 8
2Value of each term: 28, −7, 8
3Add the terms: 28 + (−7) + 8 = 29
28 + −7 + 8 = 29
✅ Answer: 29
๐ Page 34Figure it Out · Q1(b)
Find the value of 39 − 2 × 6 + 11 by writing its terms.
1Terms: 39, −2 × 6, 11
2Value of each term: 39, −12, 11
3Add the terms: 39 + (−12) + 11 = 38
39 + −2 × 6 + 11 = 38
✅ Answer: 38
๐ Page 34Figure it Out · Q1(c)
Find the value of 40 − 10 + 10 + 10 by writing its terms.
1Terms: 40, −10, 10, 10
2Value of each term: 40, −10, 10, 10
3Add the terms: 40 + (−10) + 10 + 10 = 50
40 + −10 + 10 + 10 = 50
✅ Answer: 50
๐ Page 34Figure it Out · Q1(d)
Find the value of 48 − 10 × 2 + 16 ÷ 2 by writing its terms.
1Terms: 48, −10 × 2, 16 ÷ 2
2Value of each term: 48, −20, 8
3Add the terms: 48 + (−20) + 8 = 36
48 + −10 × 2 + 16 ÷ 2 = 36
✅ Answer: 36
๐ Page 34Figure it Out · Q1(e)
Find the value of 6 × 3 − 4 × 8 × 5 by writing its terms.
1Terms: 6 × 3, −4 × 8 × 5
2Value of each term: 18, −160
3Add the terms: 18 + (−160) = −142
6 × 3 + −4 × 8 × 5 = −142
✅ Answer: −142
๐ Page 34Figure it Out · Q2(a)
Write a story for 89 + 21 − 10 and find its value.
1Story: A shelf has 89 books. The librarian adds 21 new books. Then 10 books are borrowed.
289 + 21 = 110; 110 − 10 = 100
✅ Answer: 100 books are on the shelf.
๐ Page 34Figure it Out · Q2(b)
Write a story for 5 × 12 − 6 and find its value.
1Story: Meena buys 5 boxes with 12 bangles each. 6 bangles break.
25 × 12 = 60; 60 − 6 = 54
✅ Answer: 54 bangles are left.
๐ Page 34Figure it Out · Q2(c)
Write a story for 4 × 9 + 2 × 6 and find its value.
1Story: 4 benches seat 9 children each and 2 smaller benches seat 6 each.
24 × 9 = 36; 2 × 6 = 12; 36 + 12 = 48
✅ Answer: 48 children can sit.
๐ Page 34Figure it Out · Q3(a)
Elsa got 100 gold coins and doubled them. Anna got 100 coins and has half left. Write an expression for their coins together, identify the terms, and find the value.
1Elsa: 2 × 100
2Anna: 100 ÷ 2
3Together: 2 × 100 + 100 ÷ 2
4Terms: 2 × 100 and 100 ÷ 2
5200 + 50 = 250
✅ Answer: 2 × 100 + 100 ÷ 2 = 250 gold coins.
๐ Page 34Figure it Out · Q3(b)
Metro tickets: ₹40 per adult, ₹20 per child. Find the cost for (i) 4 adults and 3 children (ii) two groups of 3 adults each.
1(i) 4 × 40 + 3 × 20 → terms 4 × 40 and 3 × 20 → 160 + 60 = 220
2(ii) 2 × (3 × 40) → one term → 2 × 120 = 240
✅ Answer: (i) ₹220 (ii) ₹240
๐ Textbook Page 35
๐ Page 35Figure it Out · Q3(c)
Find the total height of the window: border 3 cm, grill 2 cm, gap 5 cm (see the picture).Window from top to bottom: border, then gap and grill again and again, last gap, border.
1Count from the picture: 2 borders (top and bottom), 6 grills and 7 gaps.
2Expression: 2 × 3 + 6 × 2 + 7 × 5
3Terms: 2 × 3, 6 × 2, 7 × 5
46 + 12 + 35 = 53
✅ Answer: 2 × 3 + 6 × 2 + 7 × 5 = 53 cm.
๐ Page 35Removing brackets I
Find 200 − (40 + 3) in two ways.
1Bracket first: 200 − 43 = 157
2Or subtract one by one: 200 − 40 = 160; 160 − 3 = 157
3So 200 − (40 + 3) = 200 − 40 − 3.
✅ Answer: 157
⚠️ Common mistake: Writing 200 − 40 + 3 = 163. Both 40 and 3 must be subtracted.
๐ Textbook Page 36
๐ Page 36Example 13
Write 500 − (250 − 100) without brackets.
1Value: 250 − 100 = 150; 500 − 150 = 350.
2500 − 250 takes away 100 too many, so add 100 back.
3500 − (250 − 100) = 500 − 250 + 100 = 350
✅ Answer: 500 − 250 + 100 = 350
๐ก Tip: A minus sign before a bracket changes the sign of EVERY term inside.
๐ Page 36Example 14
Hira has 28 coins in one bag and 35 in another. She gives away 10 from the second bag. Write the expression without brackets.
128 + (35 − 10)
2A plus sign before a bracket keeps the signs: 28 + 35 − 10
3= 53
✅ Answer: 28 + 35 − 10 = 53 coins.
๐ Textbook Page 37
๐ Page 37Tinker the Terms I
Fill the blanks with as little calculation as possible (first column is done as an example).
1−15 is 1 MORE than −16 → 53 + (−15) is 1 more than 37 = 38.
252 is 1 less than 53 → 52 + (−16) = 36. −17 is 1 less than −16 → 53 + (−17) = 36.
3−87 + (−16) = −103. −88 is 1 less but −15 is 1 more → still −103.
4−86 is 1 more, −18 is 2 less → 1 less overall → −104.
5−97 is 10 less, −26 is 10 less → 20 less → −123.
| Column 1 | Column 2 | Column 3 |
|---|---|---|
| 53 + (−16) = 37 | 53 + (−16) = 37 | −87 + (−16) = −103 |
| 54 + (−16) = 38 | 52 + (−16) = 36 | −88 + (−15) = −103 |
| 53 + (−15) = 38 | 53 + (−17) = 36 | −86 + (−18) = −104 |
| −97 + (−26) = −123 |
✅ Answer: 38; 36, 36; −103, −103, −104, −123
๐ Page 37Figure it Out · Q1(a)
Fill in the blanks and signs so both sides are equal: 24 + (6 − 4) = 24 + 6 ☐ ___
1Plus before a bracket: signs stay the same. Minus before a bracket: every sign inside changes.
2So: 24 + 6 − 4
✅ Answer: 24 + 6 − 4
๐ Page 37Figure it Out · Q1(b)
Fill in the blanks and signs so both sides are equal: 38 + (___ ☐ ___) = 38 + 9 − 4
1Plus before a bracket: signs stay the same. Minus before a bracket: every sign inside changes.
2So: 38 + (9 − 4)
✅ Answer: 38 + (9 − 4)
๐ Page 37Figure it Out · Q1(c)
Fill in the blanks and signs so both sides are equal: 24 − (6 + 4) = 24 ☐ 6 − 4
1Plus before a bracket: signs stay the same. Minus before a bracket: every sign inside changes.
2So: 24 − 6 − 4
✅ Answer: 24 − 6 − 4
๐ Page 37Figure it Out · Q1(d)
Fill in the blanks and signs so both sides are equal: 24 − 6 − 4 = 24 − 6 ☐ ___
1Plus before a bracket: signs stay the same. Minus before a bracket: every sign inside changes.
2So: 24 − 6 − 4
✅ Answer: 24 − 6 − 4
๐ Page 37Figure it Out · Q1(e)
Fill in the blanks and signs so both sides are equal: 27 − (8 + 3) = 27 ☐ 8 ☐ 3
1Plus before a bracket: signs stay the same. Minus before a bracket: every sign inside changes.
2So: 27 − 8 − 3
✅ Answer: 27 − 8 − 3
๐ Page 37Figure it Out · Q1(f)
Fill in the blanks and signs so both sides are equal: 27 − (___ ☐ ___) = 27 − 8 + 3
1Plus before a bracket: signs stay the same. Minus before a bracket: every sign inside changes.
2So: 27 − (8 − 3)
✅ Answer: 27 − (8 − 3)
๐ Page 37Figure it Out · Q2(a)
Remove the brackets: 14 + (12 + 10)
1Plus before the bracket → keep signs.
214 + (12 + 10) = 14 + 12 + 10
3Value = 36
✅ Answer: 14 + 12 + 10 = 36
๐ Page 37Figure it Out · Q2(b)
Remove the brackets: 14 − (12 + 10)
1Minus before the bracket → change every sign inside.
214 − (12 + 10) = 14 − 12 − 10
3Value = −8
✅ Answer: 14 − 12 − 10 = −8
๐ Page 37Figure it Out · Q2(c)
Remove the brackets: 14 + (12 − 10)
1Plus before the bracket → keep signs.
214 + (12 − 10) = 14 + 12 − 10
3Value = 16
✅ Answer: 14 + 12 − 10 = 16
๐ Page 37Figure it Out · Q2(d)
Remove the brackets: 14 − (12 − 10)
1Minus before the bracket → change every sign inside.
214 − (12 − 10) = 14 − 12 + 10
3Value = 12
✅ Answer: 14 − 12 + 10 = 12
๐ Page 37Figure it Out · Q2(e)
Remove the brackets: −14 + (12 − 10)
1Plus before the bracket → keep signs.
2−14 + (12 − 10) = −14 + 12 − 10
3Value = −12
✅ Answer: −14 + 12 − 10 = −12
๐ Page 37Figure it Out · Q2(f)
Remove the brackets: 14 − (−12 − 10)
1Minus before the bracket → change every sign inside.
214 − (−12 − 10) = 14 + 12 + 10
3Value = 36
✅ Answer: 14 + 12 + 10 = 36
๐ Page 37Figure it Out · Q3(a)
Guess, then find: are (6 + 10) − 2 and 6 + (10 − 2) equal?
1(6 + 10) − 2 = 14
26 + (10 − 2) = 14
3Equal ✔
✅ Answer: Equal: both are 14
๐ Page 37Figure it Out · Q3(b)
Guess, then find: are 16 − (8 − 3) and (16 − 8) − 3 equal?
116 − (8 − 3) = 11
2(16 − 8) − 3 = 5
3Not equal ✘ — in 16 − (8 − 3) the 3 is ADDED back; in (16 − 8) − 3 it is subtracted.
✅ Answer: Not equal (11 and 5)
๐ Page 37Figure it Out · Q3(c)
Guess, then find: are 27 − (18 + 4) and 27 + (−18 − 4) equal?
127 − (18 + 4) = 5
227 + (−18 − 4) = 5
3Equal ✔
✅ Answer: Equal: both are 5
๐ก Tip: Two expressions are equal when, after removing brackets, they have the same terms.
๐ Textbook Page 38
๐ Page 38Figure it Out · Q4(a)
Which have the same value? 319 + 537, 319 − 537, −537 + 319, 537 − 319
1Write the terms: (319, 537), (319, −537), (−537, 319), (537, −319).
2319 − 537 and −537 + 319 have the same terms → same value (−218).
3537 − 319 = 218 is the opposite of these; 319 + 537 = 856 is different.
✅ Answer: 319 − 537 = −537 + 319
๐ Page 38Figure it Out · Q4(b)
Which have the same value? 87 + 46 − 109 (written three times), 87 − 46 + 109, 87 − (46 + 109), (87 − 46) + 109
187 + 46 − 109 → terms 87, 46, −109 (all three copies are the same) → value 24.
287 − 46 + 109 → terms 87, −46, 109 → value 150.
3(87 − 46) + 109 = 87 − 46 + 109 → same terms → 150.
487 − (46 + 109) = 87 − 46 − 109 → terms 87, −46, −109 → value −68.
✅ Answer: The three copies of 87 + 46 − 109 are equal; 87 − 46 + 109 = (87 − 46) + 109; 87 − (46 + 109) is different.
๐ Page 38Figure it Out · Q5(a)
Add brackets so that: 34 − 9 + 12 = 13
1Try putting a bracket after a minus sign — it changes the signs inside.
234 − (9 + 12) = 34 − 21 = 13
✅ Answer: 34 − (9 + 12)
๐ Page 38Figure it Out · Q5(b)
Add brackets so that: 56 − 14 − 8 = 34
1Try putting a bracket after a minus sign — it changes the signs inside.
2(56 − 14) − 8 = 42 − 8 = 34
✅ Answer: (56 − 14) − 8
๐ก Tip: In (b) the value is already 34 without brackets; (56 − 14) − 8 keeps it the same.
๐ Page 38Figure it Out · Q5(c)
Add brackets so that: −22 − 12 + 10 + 22 = −22
1Try putting a bracket after a minus sign — it changes the signs inside.
2−22 − (12 + 10) + 22 = −22 − 22 + 22 = −22
✅ Answer: −22 − (12 + 10) + 22
๐ Page 38Figure it Out · Q6(a)
Fill the blanks using reasoning: 423 + ___ = 419 + ___
1423 is 4 more than 419.
2So the blank on the right must be 4 more than the blank on the left.
3Example: 423 + 10 = 419 + 14 (both 433).
✅ Answer: 423 + 10 = 419 + 14 (any pair where the right blank is 4 more)
๐ Page 38Figure it Out · Q6(b)
Fill the blank: 207 − 68 = 210 − ___
1210 is 3 more than 207.
2To keep the answer the same, take away 3 more: 68 + 3 = 71.
3Check: 207 − 68 = 139 and 210 − 71 = 139 ✔
✅ Answer: 71
๐ Page 38Figure it Out · Q7
Using 2, 3 and 5 (each once) with +, − and brackets, make as many different values as possible.
1The first number is added; each of the other two is added or subtracted (brackets only change signs).
22 + 3 + 5 = 10
33 − 2 + 5 = 6
42 − 3 + 5 = 4
52 + 3 − 5 = 0
63 − 2 − 5 = −4
72 − (3 + 5) = −6
✅ Answer: Six values: 10, 6, 4, 0, −4, −6.
๐ Page 38Figure it Out · Q8
Jasoda subtracts 9 by subtracting 10 and adding 1 (36 − 9 = 26 + 1). (a) Is she always right? Why? (b) Give other similar strategies.
1(a) 9 = 10 − 1, so n − 9 = n − (10 − 1) = n − 10 + 1. Yes, always correct.
2(b) Subtract 8: subtract 10, add 2 (55 − 8 = 45 + 2 = 47).
3Add 9: add 10, subtract 1 (47 + 9 = 57 − 1 = 56).
4Subtract 99: subtract 100, add 1 (345 − 99 = 245 + 1 = 246).
✅ Answer: (a) Yes, because −9 = −10 + 1. (b) e.g. −8 = −10 + 2, +9 = +10 − 1, −99 = −100 + 1.
๐ Page 38Figure it Out · Q9
Which of (a) 73 − (14 + 1) (b) 73 − (14 − 1) (c) 73 + (−14 + 1) (d) 73 + (−14 − 1) equal 73 − 14 + 1, and which equal 73 − 14 − 1?
1(a) 73 − 14 − 1
2(b) 73 − 14 + 1
3(c) 73 − 14 + 1
4(d) 73 − 14 − 1
✅ Answer: 73 − 14 + 1 = (b) and (c); 73 − 14 − 1 = (a) and (d).
๐ Page 38Example 15
Lhamo and Norbu each have a cutlet (₹43) and a rasgulla (₹24). Write the total. If Sangmu joins with the same order?
1One share = 43 + 24.
2Two shares = 2 × (43 + 24) = 2 × 43 + 2 × 24 = 134.
3With Sangmu: 3 × (43 + 24) = 3 × 67 = 201.
✅ Answer: 2 × (43 + 24) = ₹134; with Sangmu 3 × (43 + 24) = ₹201.
⚠️ Common mistake: 2 × 43 + 24 means only one rasgulla — use brackets.
๐ Textbook Page 40
๐ Page 40Explain
Why is 5 × 4 + 3 ≠ 5 × (4 + 3)? Is 5 × (4 + 3) = 5 × (3 + 4) = (3 + 4) × 5?
15 × 4 + 3 = 20 + 3 = 23 (3 is added once).
25 × (4 + 3) = 5 × 7 = 35 (3 is added 5 times).
35 × (4 + 3), 5 × (3 + 4) and (3 + 4) × 5 are all 5 × 7 = 35.
✅ Answer: 23 ≠ 35. Yes, the other three are all equal to 35.
๐ Textbook Page 41
๐ Page 41Example 18
Find 97 × 25 quickly.
197 × 25 = (100 − 3) × 25
2= 100 × 25 − 3 × 25
3= 2500 − 75 = 2425
✅ Answer: 2425
๐ Page 41Use the method (a)
Find 95 × 8 using the distributive property.
1(100 − 5) × 8
2= 800 − 40
3= 760
✅ Answer: 760
๐ Page 41Use the method (b)
Find 104 × 15 using the distributive property.
1(100 + 4) × 15
2= 1500 + 60
3= 1560
✅ Answer: 1560
๐ Page 41Use the method (c)
Find 49 × 50 using the distributive property.
1(50 − 1) × 50
2= 2500 − 50
3= 2450
✅ Answer: 2450
๐ Page 41Math Talk
Is this method quicker? Which other products are quick to find this way?
1It is quicker when one number is close to 10, 50, 100, 1000, …
298 × 7 = 700 − 14 = 686
3103 × 9 = 900 + 27 = 927
4199 × 4 = 800 − 4 = 796
✅ Answer: Yes — for numbers near a round number, e.g. 98 × 7, 103 × 9, 199 × 4.
๐ Page 41Figure it Out · Q1
Fill the blanks and signs so that both sides are equal ((a) and (b) are already complete).
1Distributive property: a × (b + c) = a × b + a × c and a × (b − c) = a × b − a × c.
2The multiplier outside the bracket multiplies EACH term inside.
3The table gives all the answers.
| Question | Answer | |
|---|---|---|
| (c) | 3 × (5 + 8) = 3 × 5 □ 3 × ___ | 3 × 5 + 3 × 8 |
| (d) | (9 + 2) × 4 = 9 × 4 □ 2 × ___ | 9 × 4 + 2 × 4 |
| (e) | 3 × (___ + 4) = 3 □ ___ + ___ | 3 × (10 + 4) = 3 × 10 + 3 × 4 (any number in place of 10) |
| (f) | (___ + 6) × 4 = 13 × 4 + ___ | (13 + 6) × 4 = 13 × 4 + 6 × 4 |
| (g) | 3 × (___ + ___) = 3 × 5 + 3 × 2 | 3 × (5 + 2) |
| (h) | (___ + ___) × ___ = 2 × 4 + 3 × 4 | (2 + 3) × 4 |
| (i) | 5 × (9 − 2) = 5 × 9 − 5 × ___ | 5 × 2 |
| (j) | (5 − 2) × 7 = 5 × 7 − 2 × ___ | 2 × 7 |
| (k) | 5 × (8 − 3) = 5 × 8 □ 5 × ___ | 5 × 8 − 5 × 3 |
| (l) | (8 − 3) × 7 = 8 × 7 □ 3 × 7 | 8 × 7 − 3 × 7 |
| (m) | 5 × (12 − ___) = ___ □ 5 × ___ | 5 × (12 − 3) = 5 × 12 − 5 × 3 (any number in place of 3) |
| (n) | (15 − ___) × 7 = ___ □ 6 × 7 | (15 − 6) × 7 = 15 × 7 − 6 × 7 |
| (o) | 5 × (___ − ___) = 5 × 9 − 5 × 4 | 5 × (9 − 4) |
| (p) | (___ − ___) × ___ = 17 × 7 − 9 × 7 | (17 − 9) × 7 |
✅ Answer: See the table.
๐ Textbook Page 42
๐ Page 42Figure it Out · Q2(a)
Fill <, > or = by reasoning: (8 − 3) × 29 ☐ (3 − 8) × 29
18 − 3 = 5 is positive and 3 − 8 = −5 is negative, both times 29.
2(Check: 145 and −145)
✅ Answer: (8 − 3) × 29 > (3 − 8) × 29
๐ Page 42Figure it Out · Q2(b)
Fill <, > or = by reasoning: 15 + 9 × 18 ☐ (15 + 9) × 18
1Left adds 15 once; right adds 15 eighteen times.
2(Check: 177 and 432)
✅ Answer: 15 + 9 × 18 < (15 + 9) × 18
๐ Page 42Figure it Out · Q2(c)
Fill <, > or = by reasoning: 23 × (17 − 9) ☐ 23 × 17 + 23 × 9
1Left is 23 × 17 − 23 × 9; right ADDS 23 × 9.
2(Check: 184 and 598)
✅ Answer: 23 × (17 − 9) < 23 × 17 + 23 × 9
๐ Page 42Figure it Out · Q2(d)
Fill <, > or = by reasoning: (34 − 28) × 42 ☐ 34 × 42 − 28 × 42
1Distributive property: exactly the same.
2(Check: 252 and 252)
✅ Answer: (34 − 28) × 42 = 34 × 42 − 28 × 42
๐ Page 42Figure it Out · Q3
2 × (1 + 6) = 14. Find other ways: ___ × (___ + ___) = 14.
114 = 2 × 7 or 7 × 2 or 1 × 14.
22 × 7: 2 × (2 + 5), 2 × (3 + 4), 2 × (0 + 7)
37 × 2: 7 × (1 + 1), 7 × (0 + 2)
41 × 14: 1 × (6 + 8), 1 × (9 + 5) …
✅ Answer: 2 × (2 + 5), 2 × (3 + 4), 7 × (1 + 1), 1 × (6 + 8)
๐ Page 42Figure it Out · Q4
Find the sum of the numbers in each picture in at least two ways, and write the expressions.
1Picture I: five 4s and four 8s → 5 × 4 + 4 × 8 = 20 + 32 = 52.
2Or row by row: (4 + 8 + 4) + (8 + 4 + 8) + (4 + 8 + 4) = 16 + 20 + 16 = 52.
3Picture II: eight 5s and eight 6s → 8 × 5 + 8 × 6 = 8 × (5 + 6) = 88.
4Or each row adds to 22: 4 × 22 = 88.
✅ Answer: Picture I = 52; Picture II = 88
๐ Page 42Figure it Out · Q1(a)
Rahim supplies 9 kg and Shyam 11 kg of mangoes every day, 7 days a week. Find the total in a week.
1Per day: 9 + 11
2Week: 7 × (9 + 11)
3= 7 × 20 = 140
✅ Answer: 7 × (9 + 11) = 140 kg
๐ Textbook Page 43
๐ Page 43Figure it Out · Q1(b)
Binu earns ₹20,000 a month and spends ₹5,000 + ₹5,000 + ₹2,000 every month. How much does she save in a year?
1Saving per month: 20,000 − (5,000 + 5,000 + 2,000) = 8,000
2Year: 12 × [20,000 − (5,000 + 5,000 + 2,000)]
3= 12 × 8,000 = 96,000
✅ Answer: ₹96,000
๐ Page 43Figure it Out · Q1(c)
A snail climbs 3 cm each day and slips 2 cm each night. The post is 10 cm high. On which day does it reach the top?
1Each full day and night: 3 − 2 = 1 cm up.
2After 7 days and nights: 7 × (3 − 2) = 7 cm.
3On the 8th day it climbs 3 cm: 7 + 3 = 10 cm — it reaches the top before slipping.
✅ Answer: On the 8th day.
⚠️ Common mistake: Answering 10 days (10 ÷ 1). On the last day the snail does not slip back.
๐ Page 43Figure it Out · Q2
Melvin reads a two-page story every day except Tuesdays and Saturdays. How many stories in 8 weeks? Which expressions describe this?
1Reading days per week = 7 − 2 = 5 → one story a day.
2Stories in 8 weeks = (7 − 2) × 8 = 40.
37 × 8 − 2 × 8 = 56 − 16 = 40 also works.
45 × 2 × 8 = 80 counts PAGES, not stories.
✅ Answer: 40 stories — expressions (b) (7 − 2) × 8 and (g) 7 × 8 − 2 × 8.
๐ Page 43Figure it Out · Q3(a)
Find 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 10 in different ways.
1Way 1: pairs (1 − 2) + (3 − 4) + … + (9 − 10) = 5 × (−1) = −5
2Way 2: (1 + 3 + 5 + 7 + 9) − (2 + 4 + 6 + 8 + 10) = 25 − 30 = −5
✅ Answer: −5
๐ Page 43Figure it Out · Q3(b)
Find 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 in different ways.
1Way 1: five pairs (1 − 1) = 0 each → 0
2Way 2: five 1s minus five 1s = 5 − 5 = 0
✅ Answer: 0
๐ Textbook Page 44
๐ Page 44Figure it Out · Q4(a)
Compare: 49 − 7 + 8 ☐ 49 − 7 + 8
1Same expression.
2(Check: 50 and 50)
✅ Answer: 49 − 7 + 8 = 49 − 7 + 8
๐ Page 44Figure it Out · Q4(b)
Compare: 83 × 42 − 18 ☐ 83 × 40 − 18
183 × 42 is more than 83 × 40.
2(Check: 3468 and 3302)
✅ Answer: 83 × 42 − 18 > 83 × 40 − 18
๐ Page 44Figure it Out · Q4(c)
Compare: 145 − 17 × 8 ☐ 145 − 17 × 6
1Left takes away more (17 × 8).
2(Check: 9 and 43)
✅ Answer: 145 − 17 × 8 < 145 − 17 × 6
๐ Page 44Figure it Out · Q4(d)
Compare: 23 × 48 − 35 ☐ 23 × (48 − 35)
1Right is 23 × 13, much smaller than 23 × 48 − 35.
2(Check: 1069 and 299)
✅ Answer: 23 × 48 − 35 > 23 × (48 − 35)
๐ Page 44Figure it Out · Q4(e)
Compare: (16 − 11) × 12 ☐ −11 × 12 + 16 × 12
1Distributive property and swapping terms.
2(Check: 60 and 60)
✅ Answer: (16 − 11) × 12 = −11 × 12 + 16 × 12
๐ Page 44Figure it Out · Q4(f)
Compare: (76 − 53) × 88 ☐ 88 × (53 − 76)
1Left is 23 × 88 (positive), right is −23 × 88.
2(Check: 2024 and −2024)
✅ Answer: (76 − 53) × 88 > 88 × (53 − 76)
๐ Page 44Figure it Out · Q4(g)
Compare: 25 × (42 + 16) ☐ 25 × (43 + 15)
142 + 16 = 43 + 15 = 58.
2(Check: 1450 and 1450)
✅ Answer: 25 × (42 + 16) = 25 × (43 + 15)
๐ Page 44Figure it Out · Q4(h)
Compare: 36 × (28 − 16) ☐ 35 × (27 − 15)
1Both brackets are 12; 36 > 35.
2(Check: 432 and 420)
✅ Answer: 36 × (28 − 16) > 35 × (27 − 15)
๐ Page 44Figure it Out · Q5(a)
Which are equal to 83 − 37 − 12? (i) 84 − 38 − 12 (ii) 84 − (37 + 12) (iii) 83 − 38 − 13 (iv) −37 + 83 − 12
1(i) 84 is 1 more and 38 is 1 more → no change ✔
2(ii) = 84 − 37 − 12 → 1 more ✘
3(iii) 38 and 13 are each 1 more → 2 less ✘
4(iv) same terms in a different order ✔
✅ Answer: (i) and (iv)
๐ Page 44Figure it Out · Q5(b)
Which are equal to 93 + 37 × 44 + 76? (i) 37 + 93 × 44 + 76 (ii) 93 + 37 × 76 + 44 (iii) (93 + 37) × (44 + 76) (iv) 37 × 44 + 93 + 76
1Terms of the given expression: 93, 37 × 44, 76.
2(iv) has exactly these terms ✔
3(i), (ii), (iii) have different terms ✘
✅ Answer: Only (iv)
๐ Page 44Figure it Out · Q6
Choose a number and create ten different expressions with that value.
1Example: 26
210 + 16, 30 − 4, 13 × 2, 52 ÷ 2, 5 + 3 × 7
35 × 5 + 1, 2 × (10 + 3), 100 − 74, 6 × 4 + 2, 3 × 9 − 1
✅ Answer: Ten expressions for 26 (as above).
๐ Textbook Page 45
๐ Page 45Expression Engineer
Using four 4s and +, −, ×, ÷ and brackets, make every number from 1 to 20.
1You may join two 4s to make 44.
2With only +, −, ×, ÷ we can make 1–10, 12, 15, 16, 17 and 20.
311, 13, 14, 18 and 19 cannot be made this way — they need a decimal point (.4 = 0.4) or a square root (√4 = 2). These are marked *.
| Value | Four 4s |
|---|---|
| 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 |
| 11 | 4 ÷ .4 + 4 ÷ 4 * |
| 12 | (44 + 4) ÷ 4 |
| 13 | 44 ÷ 4 + √4 * |
| 14 | 4 + 4 + 4 + √4 * |
| 15 | 44 ÷ 4 + 4 |
| 16 | 4 + 4 + 4 + 4 |
| 17 | 4 × 4 + 4 ÷ 4 |
| 18 | 4 × 4 + 4 − √4 * |
| 19 | (4 + 4 − .4) ÷ .4 * |
| 20 | (4 + 4 ÷ 4) × 4 |
✅ Answer: See the table.
๐ Page 45Expression Engineer
Using 1, 2, 3, 4 and 5 exactly once, get as many values as possible between −10 and +10.
1Change the signs and use one × to move the value up or down.
2Every whole number from −10 to 10 is possible — the table shows one way for each.
| Value | Expression |
|---|---|
| −10 | 1 × 2 − 3 − 4 − 5 |
| −9 | 1 + 2 − 3 − 4 − 5 |
| −8 | 1 − 2 − 3 × 4 + 5 |
| −7 | 1 − 2 + 3 − 4 − 5 |
| −6 | 1 − 2 × 3 + 4 − 5 |
| −5 | 1 − 2 − 3 + 4 − 5 |
| −4 | 1 + 2 − 3 × 4 + 5 |
| −3 | 1 + 2 + 3 − 4 − 5 |
| −2 | 1 + 2 × 3 − 4 − 5 |
| −1 | 1 + 2 − 3 + 4 − 5 |
| 0 | 1 × 2 − 3 − 4 + 5 |
| 1 | 1 + 2 − 3 − 4 + 5 |
| 2 | 1 × 3 − 2 − 4 + 5 |
| 3 | 1 − 2 + 3 − 4 + 5 |
| 4 | 1 − 2 × 3 + 4 + 5 |
| 5 | 1 + 2 + 3 + 4 − 5 |
| 6 | 1 + 2 × 3 + 4 − 5 |
| 7 | 1 + 2 + 3 − 4 + 5 |
| 8 | 1 + 2 × 3 − 4 + 5 |
| 9 | 1 + 2 − 3 + 4 + 5 |
| 10 | 1 + 2 + 3 × 4 − 5 |
✅ Answer: All 21 values from −10 to 10 can be made.
๐ Page 45Expression Engineer
Using the numbers 0 to 9 exactly once, make an expression with value 100.
1Add the small numbers and multiply the big ones: 8 × 9 = 72.
20 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
328 + 72 = 100.
✅ Answer: 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9 = 100
๐ Where do we use expressions?
- Shopping bills and change, saving money every month, tickets for a group.
- Quick mental maths: 97 × 25 = 2500 − 75.
๐ Link to higher classes
- Class 7 Ch 4: expressions with letter-numbers (algebra).
- Class 8: We Distribute, Yet Things Multiply — (a + b)², (a − b)², (a + b)(a − b).
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