Figure it Out
Q1. Fill in the blanks to make the expressions equal on both sides of the = sign: (a) 13 + 4 = ____ + 6 (b) 22 + ____ = 6 × 5 (c) 8 × ____ = 64 ÷ 2 (d) 34 – ____ = 25
Answer:
(a) 13 + 4 = 17, so we need a number that when added to 6 gives 17, which is 11. So blank = 11.
(b) 6 × 5 = 30, so 22 + blank = 30 gives blank = 8.
(c) 64 ÷ 2 = 32, so 8 × blank = 32 gives blank = 4.
(d) 34 – blank = 25 gives blank = 9.
Q2. Arrange the following expressions in ascending (increasing) order of their values. (a) 67 – 19 (b) 67 – 20 (c) 35 + 25 (d) 5 × 11 (e) 120 ÷ 3
Answer:
First, calculate each value:
(a) 67 – 19 = 48
(b) 67 – 20 = 47
(c) 35 + 25 = 60
(d) 5 × 11 = 55
(e) 120 ÷ 3 = 40
Now arrange in ascending order: 40, 47, 48, 55, 60.
So the order is: (e), (b), (a), (d), (c).
Q1. Fill in the blanks with numbers, and boxes by signs, so that the expressions on both sides are equal. (a) 3 × (6 + 7) = 3 × 6 + 3 × 7 (b) (8 + 3) × 4 = 8 × 4 + 3 × 4 (c) 3 × (5 + 8) = 3 × 5 [ ] 3 × ____ (d) (9 + 2) × 4 = 9 × 4 [ ] 2 ×____ (e) 3 × (____ + 4) = 3 ____+____ (f) (____+ 6) × 4 = 13 × 4 + ____ (g) 3 × (____+____) = 3 × 5 + 3 × 2 (h) (____+____)×____= 2 × 4 + 3 × 4 (i) 5 × (9 – 2) = 5 × 9 – 5 × ____ (j) (5 – 2) × 7 = 5 × 7 – 2 × ____ (k) 5 × (8 – 3) = 5 × 8 [ ] 5 × ____ (l) (8 – 3) × 7 = 8 × 7 [ ] 3 × 7 (m) 5 × (12 –____) =____ [ ] 5 ×____ (n) (15 –____) × 7 =____ [ ] 6 × 7 (o) 5 × (____–____) = 5 × 9 – 5 × 4 (p) (____–____) × ____= 17 × 7 – 9 × 7
Answer:
(a) 3 × (6 + 7) = 3 × 6 + 3 × 7 (already filled, 7). (b) (8 + 3) × 4 = 8 × 4 + 3 × 4 (already filled, 3). (c) 3 × (5 + 8) = 3 × 5 + 3 × 8. So fill '+' and 8. (d) (9 + 2) × 4 = 9 × 4 + 2 × 4. So fill '+' and 4. (e) 3 × (5 + 4) = 3 × 5 + 3 × 4. So fill 5, ×5, +3×4. (f) (13 + 6) × 4 = 13 × 4 + 6 × 4. So fill 13 and 6×4. (g) 3 × (5 + 2) = 3 × 5 + 3 × 2. So fill 5 and 2. (h) (2 + 3) × 4 = 2 × 4 + 3 × 4. So fill 2, 3, 4. (i) 5 × (9 – 2) = 5 × 9 – 5 × 2. So fill 2. (j) (5 – 2) × 7 = 5 × 7 – 2 × 7. So fill 7. (k) 5 × (8 – 3) = 5 × 8 – 5 × 3. So fill '-' and 3. (l) (8 – 3) × 7 = 8 × 7 – 3 × 7. So fill '-' . (m) 5 × (12 – 2) = 5 × 12 – 5 × 2. So fill 2, -, 12, 2. (n) (15 – 6) × 7 = 15 × 7 – 6 × 7. So fill 6, 15×7, -. (o) 5 × (9 – 4) = 5 × 9 – 5 × 4. So fill 9 and 4. (p) (17 – 9) × 7 = 17 × 7 – 9 × 7. So fill 17, 9, 7.
Q2. In the boxes below, fill ‘<’, ‘>’ or ‘=’ after analysing the expressions on the LHS and RHS. Use reasoning and understanding of terms and brackets to figure this out and not by evaluating the expressions. (a) (8 – 3) × 29 [ ] (3 – 8) × 29 (b) 15 + 9 × 18 [ ] (15 + 9) × 18 (c) 23 × (17 – 9) [ ] 23 × 17 + 23 × 9 (d) (34 – 28) × 42 [ ] 34 × 42 – 28 × 42
Answer:
(a) (8 – 3) × 29 = 5 × 29, (3 – 8) × 29 = (-5) × 29, negative, so (8-3)×29 > (3-8)×29. Fill '>'. (b) 15 + 9 × 18 = 15 + 162 = 177, (15 + 9) × 18 = 24 × 18 = 432, so 15 + 9 × 18 < (15+9)×18. Fill '<'. (c) 23 × (17 – 9) = 23 × 8, while 23 × 17 + 23 × 9 = 23 × (17+9) = 23 × 26, so 23 × (17-9) < 23 × 17 + 23 × 9. Fill '<'. (d) (34 – 28) × 42 = 6 × 42, and 34 × 42 – 28 × 42 = (34-28)×42 = 6 × 42, so equal. Fill '='.
Q3. Here is one way to make 14: _2_ × ( _1_ + _6_ ) = 14. Are there other ways of getting 14? Fill them out below: (a) _____× (_____+_____) = 14 (b) _____× (_____+_____) = 14 (c) _____× (_____+_____) = 14 (d) _____× (_____+_____) = 14
Answer:
(a) 7 × (1 + 1) = 7 × 2 = 14. (b) 1 × (7 + 7) = 1 × 14 = 14. (c) 14 × (1 + 0) = 14 × 1 = 14. (d) 2 × (4 + 3) = 2 × 7 = 14.
Q4. Find out the sum of the numbers given in each picture below in at least two different ways. Describe how you solved it through expressions. (Note: The text does not include the actual pictures; assume a typical problem with groups of numbers.)
Answer:
Since the pictures are not provided in the text, we describe a general method: For a set of numbers arranged in rows, sum can be found by adding all numbers directly, or by grouping (e.g., adding rows first then summing totals). For example, if there are two rows of 3 and 4 apples, sum = 3+4 = 7, or count all apples. Without specific numbers, a precise solution cannot be given.
Q1. Read the situations given below. Write appropriate expressions for each of them and find their values. (a) The district market in Begur operates on all seven days of a week. Rahim supplies 9 kg of mangoes each day from his orchard and Shyam supplies 11 kg of mangoes each day from his orchard to this market. Find the amount of mangoes supplied by them in a week to the local district market. (b) Binu earns ₹20,000 per month. She spends ₹5,000 on rent, ₹5,000 on food, and ₹2,000 on other expenses every month. What is the amount Binu will save by the end of a year? (c) During the daytime a snail climbs 3 cm up a post, and during the night while asleep, accidentally slips down by 2 cm. The post is 10 cm high, and a delicious treat is on its top. In how many days will the snail get the treat?
Answer:
(a) Total per day = 9 + 11 = 20 kg. In a week (7 days): 20 × 7 = 140 kg. Expression: (9 + 11) × 7 or 7 × 9 + 7 × 11. (b) Monthly savings = 20000 - (5000 + 5000 + 2000) = 20000 - 12000 = 8000. Yearly savings = 8000 × 12 = 96000. Expression: (20000 - (5000+5000+2000)) × 12. (c) Net climb per day = 3 - 2 = 1 cm. After 7 days, snail climbs 7 cm. On day 8, it climbs 3 cm to reach 10 cm and gets the treat without slipping. So total days = 8. Expression: Snail gets treat on 8th day. (Day 7: slips to 6 cm, day 8 climbs to 9? Actually careful: Day1 climb to 3, slip to 1. Day2 climb to 4, slip to 2. Day3: 5→3, Day4:6→4, Day5:7→5, Day6:8→6, Day7:9→7, Day8: climbs to 10. So 8 days.)
Q2. Melvin reads a two-page story every day except on Tuesdays and Saturdays. How many stories would he complete reading in 8 weeks? Which of the expressions below describes this scenario? (a) 5 × 2 × 8 (b) (7 – 2) × 8 (c) 8 × 7 (d) 7 × 2 × 8 (e) 7 × 5 – 2 (f) (7 + 2) × 8 (g) 7 × 8 – 2 × 8 (h) (7 – 5) × 8
Answer:
He reads 5 days a week (excluding Tuesday and Saturday). Each day he reads 2 pages, so in a week he reads 5 × 2 = 10 pages. In 8 weeks, 10 × 8 = 80 pages. Since each story is 2 pages, number of stories = 80 ÷ 2 = 40. The expression that gives 40 is (a) 5 × 2 × 8 = 80? Wait that gives pages, not stories. Actually 5 × 2 × 8 = 80 pages, but stories = pages/2 = 40. None of the options directly give 40? Let's check: (a) 5×2×8 = 80 (pages) but if each story is 2 pages, then stories = (5×2×8)/2 = 5×8 = 40. So (a) represents pages, but if interpreted as stories per week? The problem says 'two-page story', so reading 2 pages per day means 1 story per day? Actually it says 'a two-page story every day' meaning each day he reads one story of 2 pages. So in 5 days a week, he reads 5 stories. In 8 weeks, 5×8 = 40 stories. Expression: 5 × 8. Among options, (b) (7-2)×8 = 5×8 = 40, and (g) 7×8 - 2×8 = 56-16=40. So (b) and (g) are correct. (a) gives 80, which is pages, not stories. So answer: (b) and (g).
Q3. Find different ways of evaluating the following expressions: (a) 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + 9 – 10 (b) 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1
Answer:
(a) Pair them: (1-2) = -1, (3-4) = -1, (5-6) = -1, (7-8) = -1, (9-10) = -1, total = -5. Another way: group positive: 1+3+5+7+9 = 25, negatives: 2+4+6+8+10 = 30, difference = 25-30 = -5. (b) Pair: (1-1) = 0 for each pair, 5 pairs, total = 0. Another way: There are 5 ones and 5 negative ones, sum = 0.
Q4. Compare the following pairs of expressions using ‘<’, ‘>’ or ‘=’ or by reasoning. (a) 49 – 7 + 8 [ ] 49 – 7 + 8 (b) 83 × 42 – 18 [ ] 83 × 40 – 18 (c) 145 – 17 × 8 [ ] 145 – 17 × 6 (d) 23 × 48 – 35 [ ] 23 × (48 – 35) (e) (16 – 11) × 12 [ ] –11 × 12 + 16 × 12 (f) (76 – 53) × 88 [ ] 88 × (53 – 76) (g) 25 × (42 + 16) [ ] 25 × (43 + 15) (h) 36 × (28 – 16) [ ] 35 × (27 – 15)
Answer:
(a) Same expression, so '='. (b) LHS: 83×42 – 18, RHS: 83×40 – 18. Since 83×42 > 83×40, LHS > RHS. (c) LHS: 145 – 17×8 = 145 – 136 = 9, RHS: 145 – 17×6 = 145 – 102 = 43, so LHS < RHS. (d) LHS: 23×48 – 35, RHS: 23×48 – 23×35 = 23×48 – 805? Actually RHS = 23×(48-35) = 23×13 = 299, LHS = 1104 - 35 = 1069, so LHS > RHS. (e) RHS: -11×12 + 16×12 = (-11+16)×12 = 5×12 = 60, LHS: (16-11)×12 = 5×12 = 60, so '='. (f) LHS: (76-53)×88 = 23×88, RHS: 88×(53-76) = 88×(-23) = -23×88, so LHS > RHS (positive vs negative). (g) LHS: 25×(42+16) = 25×58 = 1450, RHS: 25×(43+15) = 25×58 = 1450, so '='. (h) LHS: 36×(28-16) = 36×12 = 432, RHS: 35×(27-15) = 35×12 = 420, so LHS > RHS.
Q5. Identify which of the following expressions are equal to the given expression without computation. You may rewrite the expressions using terms or removing brackets. There can be more than one expression which is equal to the given expression. (a) 83 – 37 – 12 (i) 84 – 38 – 12 (ii) 84 – (37 + 12) (iii) 83 – 38 – 13 (iv) – 37 + 83 –12 (b) 93 + 37 × 44 + 76 (i) 37 + 93 × 44 + 76 (ii) 93 + 37 × 76 + 44 (iii) (93 + 37) × (44 + 76) (iv) 37 × 44 + 93 + 76
Answer:
(a) Original: 83 – 37 – 12 = 83 – (37+12) = 83 – 49 = 34. (i) 84 – 38 – 12 = 34? 84-50=34, yes. (ii) 84 – (37+12) = 84-49=35, not equal. (iii) 83 – 38 – 13 = 83-51=32, not equal. (iv) –37 + 83 –12 = 83-37-12 = 34, equal. So equal: (i) and (iv). (b) Original: 93 + 37×44 + 76. (i) 37 + 93×44 + 76 = 37 + 4092 + 76 = 4205, original = 93 + 1628 + 76 = 1797, not equal. (ii) 93 + 37×76 + 44 = 93 + 2812 + 44 = 2949, not equal. (iii) (93+37)×(44+76) = 130×120 = 15600, not equal. (iv) 37×44 + 93 + 76 = 1628 + 93 + 76 = 1797, equal. So only (iv).
Q6. Choose a number and create ten different expressions having that value.
Answer:
Let us choose the number 12. Ten expressions: (1) 6+6, (2) 3×4, (3) 24÷2, (4) 15-3, (5) 2+2+2+2+2+2, (6) 1+11, (7) 36÷3, (8) 10+2, (9) 20-8, (10) (4+2)×2.
Try This
Q1. In the following table, some expressions are given. Complete the table. Expression | 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
Answer:
Completing the table:
Expression | 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
Q1. Will swapping terms change the sum when both terms are positive? Take some more expressions and check. Can you explain why this is happening using the Token Model of integers?
Answer:
No, swapping terms does not change the sum when both terms are positive. For example, 5 + 3 = 8 and 3 + 5 = 8. Using the Token Model, if we have 5 positive tokens and add 3 positive tokens, we get 8 positive tokens regardless of the order. This is because addition is commutative.
Q2. Will swapping terms hold when there are terms having negative numbers as well? Take some more expressions and check. Can you explain why this is happening using the Token Model of integers?
Answer:
Yes, swapping terms works for negative numbers as well. For example, (–7) + 10 = 3 and 10 + (–7) = 3. Using the Token Model, we can think of 7 negative tokens and 10 positive tokens; combining them gives 3 positive tokens regardless of order. Similarly, (–5) + (–3) = –8 and (–3) + (–5) = –8. So the commutative property holds for all integers.
Q1. Will grouping terms in different ways give the same value when there are negative numbers? Take some more expressions and check. Can you explain why this is happening using the Token Model of integers?
Answer:
Yes, grouping does not change the sum. For example, (–7) + 10 + (–11):
- First group: (–7 + 10) + (–11) = 3 + (–11) = –8
- Second group: –7 + (10 + (–11)) = –7 + (–1) = –8
Using the Token Model, we can combine tokens in any order; the total count of positive and negative tokens remains the same regardless of how we group them.
Q1. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also. Can you explain why this is happening using the Token Model of integers?
Answer:
Yes, adding terms in any order gives the same value. For example, 1 + 2 + 3 + 4 = 10, and 4 + 1 + 3 + 2 = 10. With negative numbers: (–2) + 5 + (–3) + 1 = 1, and 1 + (–3) + 5 + (–2) = 1. Using the Token Model, we are simply counting the total number of positive and negative tokens; the order of counting does not affect the total.