Question 1(a)
Fill the blank: 13+4 = □+6.
Steps only
17 = □+6; □ = 17−6 = 11
Answer
11
CLASS 7 · GANITA PRAKASH PART I · CHAPTER 2
Arithmetic Expressions
Short working and clear explanations for the chapter prompts, investigations and Figure it Out questions.
125 worked answers
125 worked answers · short steps first, explanations when you need them
Figure it Out — Expressions and values
Textbook pages 25
Question 1(b)
Fill the blank: 22+□ = 6×5.
Steps only
22+□ = 30; □ = 30−22 = 8
Answer
8
Question 1(c)
Fill the blank: 8×□ = 64÷2.
Steps only
8×□ = 32; □ = 32÷8 = 4
Answer
4
Question 1(d)
Fill the blank: 34−□ = 25.
Steps only
□ = 34−25 = 9
Answer
9
Question 2
Arrange 67−19, 67−20, 35+25, 5×11 and 120÷3 in ascending order.
Steps only
a=48; b=47; c=60; d=55; e=40
40 < 47 < 48 < 55 < 60
Answer
120÷3 < 67−20 < 67−19 < 5×11 < 35+25.
Figure it Out — Terms and situations
Textbook pages 34–35
Question 1(a)
List the terms and evaluate 28−7+8.
Steps only
Terms: 28, −7, +8.
28−7+8 = 21+8 = 29
Answer
29
Question 1(b)
List the terms and evaluate 39−2×6+11.
Steps only
Terms: 39, −2×6, +11.
39−12+11 = 27+11 = 38
Answer
38
Question 1(c)
List the terms and evaluate 40−10+10+10.
Steps only
Terms: 40, −10, +10, +10.
40−10+10+10 = 40+10 = 50
Answer
50
Question 1(d)
List the terms and evaluate 48−10×2+16÷2.
Steps only
Terms: 48, −10×2, +16÷2.
48−20+8 = 28+8 = 36
Answer
36
Question 1(e)
List the terms and evaluate 6×3−4×8×5.
Steps only
Terms: 6×3, −4×8×5.
18−160 = −142
Answer
−142
Question 2(a)
Give a situation for 89+21−10 and evaluate it.
Steps only
A library has 89 books, receives 21, and lends out 10. How many remain?
89+21−10 = 100
Answer
100 books
Question 2(b)
Give a situation for 5×12−6 and evaluate it.
Steps only
Five boxes hold 12 pencils each. Six pencils are given away. How many remain?
60−6 = 54
Answer
54 pencils
Question 2(c)
Give a situation for 4×9+2×6 and evaluate it.
Steps only
Four bags hold 9 marbles each and two bags hold 6 each. How many marbles are there?
36+12 = 48
Answer
48 marbles
Question 3(a)
Two princesses each begin with 100 coins. One doubles her coins and the other keeps half. How many coins do they have together?
Steps only
terms: 2×100 and 100÷2
2×100+100÷2 = 200+50 = 250
Answer
250 coins.
Question 3(b)(i)
Adult tickets cost ₹40 and child tickets ₹20. Find the cost for 4 adults and 3 children.
Steps only
4×40+3×20 = 220
Answer
₹220
Question 3(b)(ii)
Adult tickets cost ₹40 and child tickets ₹20. Find the cost for 2 groups of 3 adults.
Steps only
2×3×40 = 240
Answer
₹240
Question 3(c)
Find the height of the textbook window: borders are 3 cm thick, each grill 2 cm thick and each gap 5 cm high.
Steps only
Two borders, six grills and seven gaps.
height = 2×3+6×2+7×5 = 6+12+35 = 53
Answer
53 cm.
Figure it Out — Brackets and signed terms
Textbook pages 37–38
Question 1(a)
Complete the equality for 24+(6−4).
Steps only
24+6−4
Answer
24+6−4
Question 1(b)
Complete the equality for 38+(□ □ □) = 38+9−4.
Steps only
38+(9−4) = 38+9−4
Answer
38+(9−4) = 38+9−4
Question 1(c)
Complete the equality for 24−(6+4).
Steps only
24−6−4
Answer
24−6−4
Question 1(d)
Complete the equality for 24−6−4 = 24−6 □ □.
Steps only
24−6−4 = 24−6−4
Answer
24−6−4 = 24−6−4
Question 1(e)
Complete the equality for 27−(8+3).
Steps only
27−8−3
Answer
27−8−3
Question 1(f)
Complete the equality for 27−(□ □ □) = 27−8+3.
Steps only
27−(8−3) = 27−8+3
Answer
27−(8−3) = 27−8+3
Question 2(a)
Remove the brackets from 14+(12+10).
Steps only
14+(12+10) = 14+12+10
Answer
14+12+10
Question 2(b)
Remove the brackets from 14−(12+10).
Steps only
14−(12+10) = 14−12−10
Answer
14−12−10
Question 2(c)
Remove the brackets from 14+(12−10).
Steps only
14+(12−10) = 14+12−10
Answer
14+12−10
Question 2(d)
Remove the brackets from 14−(12−10).
Steps only
14−(12−10) = 14−12+10
Answer
14−12+10
Question 2(e)
Remove the brackets from −14+(12−10).
Steps only
−14+(12−10) = −14+12−10
Answer
−14+12−10
Question 2(f)
Remove the brackets from 14−(−12−10).
Steps only
14−(−12−10) = 14+12+10
Answer
14+12+10
Question 3(a)
Compare (6+10)−2 and 6+(10−2).
Steps only
16−2 = 14; 6+8 = 14
Answer
Equal; both are 14.
Question 3(b)
Compare 16−(8−3) and (16−8)−3.
Steps only
16−5 = 11; 8−3 = 5
Answer
Not equal: 11 and 5.
Question 3(c)
Compare 27−(18+4) and 27+(−18−4).
Steps only
27−22 = 5; 27+(−22) = 5
Answer
Equal; both are 5.
Question 4(a)
Which expressions match among 319+537, 319−537, −537+319 and 537−319?
Steps only
319−537 = −537+319
Answer
The second and third expressions are equal.
Question 4(b)
Identify matching expressions in the six-entry set beginning 87+46−109.
Steps only
entries 1,2,3: +87+46−109
entries 4,6: +87−46+109
entry 5: +87−46−109
Answer
Entries 1, 2 and 3 match; entries 4 and 6 match.
Question 5(a)
Insert brackets into 34−9+12 to give 13.
Steps only
34−(9+12) = 13
Answer
34−(9+12) = 13
Question 5(b)
Insert brackets into 56−14−8 to give 34.
Steps only
56−(14+8) = 34
Answer
56−(14+8) = 34
Question 5(c)
Insert brackets into −22−12+10+22 to give -22.
Steps only
−22−(12+10)+22 = −22
Answer
−22−(12+10)+22 = −22
Question 6(a)
Fill 423+□ = 419+□ so that the sides are equal.
Steps only
423+0 = 419+4
Answer
One answer: 0 on the left and 4 on the right.
Question 6(b)
Fill 207−68 = 210−□.
Steps only
210 is 3 more than 207
□ = 68+3 = 71
Answer
71
Question 7
Use 2,3,5 once each with +,− and brackets to make as many values as possible.
Steps only
2−(3+5) = −6
3−(2+5) = −4
5−(2+3) = 0
2+5−3 = 4
3+5−2 = 6
2+3+5 = 10
Answer
Possible values: −6, −4, 0, 4, 6 and 10.
Question 8(a)
Does subtracting 10 and then adding 1 always subtract 9?
Steps only
n−10+1 = n−(10−1) = n−9
Answer
Yes.
Question 8(b)
Give similar methods for subtracting numbers near multiples of ten.
Steps only
n−19 = n−20+1
n−29 = n−30+1
n−11 = n−10−1
Answer
Examples: subtract 20 then add 1 for −19; subtract 10 then subtract 1 for −11.
Question 9(a)
Find matches for 73−14+1 among 73−(14+1), 73−(14−1), 73+(−14+1), 73+(−14−1).
Steps only
73−(14−1) = 73−14+1
73+(−14+1) = 73−14+1
Answer
The second and third choices.
Question 9(b)
Find matches for 73−14−1 in the same collection.
Steps only
73−(14+1) = 73−14−1
73+(−14−1) = 73−14−1
Answer
The first and fourth choices.
Figure it Out — Distributive property
Textbook pages 41–42
Question 1(a)
Complete the distributive equality in part (a).
Steps only
3(6+7) = 3×6+3×7
Answer
3(6+7) = 3×6+3×7.
Question 1(b)
Complete the distributive equality in part (b).
Steps only
(8+3)4 = 8×4+3×4
Answer
(8+3)4 = 8×4+3×4.
Question 1(c)
Complete the distributive equality in part (c).
Steps only
3(5+8) = 3×5+3×8
Answer
3(5+8) = 3×5+3×8.
Question 1(d)
Complete the distributive equality in part (d).
Steps only
(9+2)4 = 9×4+2×4
Answer
(9+2)4 = 9×4+2×4.
Question 1(e)
Complete the distributive equality in part (e).
Steps only
3(1+4) = 3×1+3×4
Answer
3(1+4) = 3×1+3×4 (one valid completion).
Question 1(f)
Complete the distributive equality in part (f).
Steps only
(13+6)4 = 13×4+6×4
Answer
(13+6)4 = 13×4+6×4.
Question 1(g)
Complete the distributive equality in part (g).
Steps only
3(5+2) = 3×5+3×2
Answer
3(5+2) = 3×5+3×2.
Question 1(h)
Complete the distributive equality in part (h).
Steps only
(2+3)4 = 2×4+3×4
Answer
(2+3)4 = 2×4+3×4.
Question 1(i)
Complete the distributive equality in part (i).
Steps only
5(9−2) = 5×9−5×2
Answer
5(9−2) = 5×9−5×2.
Question 1(j)
Complete the distributive equality in part (j).
Steps only
(5−2)7 = 5×7−2×7
Answer
(5−2)7 = 5×7−2×7.
Question 1(k)
Complete the distributive equality in part (k).
Steps only
5(8−3) = 5×8−5×3
Answer
5(8−3) = 5×8−5×3.
Question 1(l)
Complete the distributive equality in part (l).
Steps only
(8−3)7 = 8×7−3×7
Answer
(8−3)7 = 8×7−3×7.
Question 1(m)
Complete the distributive equality in part (m).
Steps only
5(12−4) = 5×12−5×4
Answer
5(12−4) = 5×12−5×4 (one valid completion).
Question 1(n)
Complete the distributive equality in part (n).
Steps only
(15−6)7 = 15×7−6×7
Answer
(15−6)7 = 15×7−6×7.
Question 1(o)
Complete the distributive equality in part (o).
Steps only
5(9−4) = 5×9−5×4
Answer
5(9−4) = 5×9−5×4.
Question 1(p)
Complete the distributive equality in part (p).
Steps only
(17−9)7 = 17×7−9×7
Answer
(17−9)7 = 17×7−9×7.
Question 2(a)
Compare (8−3)29 and (3−8)29.
Steps only
(8−3)29 > (3−8)29
Answer
>
Question 2(b)
Compare 15+9×18 and (15+9)18.
Steps only
15+9×18 < (15+9)18
Answer
<
Question 2(c)
Compare 23(17−9) and 23×17+23×9.
Steps only
23(17−9) < 23×17+23×9
Answer
<
Question 2(d)
Compare (34−28)42 and 34×42−28×42.
Steps only
(34−28)42 = 34×42−28×42
Answer
=
Question 3(a)
Give another expression of the form □×(□+□) equal to 14.
Steps only
1×(6+8) = 14
Answer
1×(6+8)
Question 3(b)
Give another expression of the form □×(□+□) equal to 14.
Steps only
2×(2+5) = 14
Answer
2×(2+5)
Question 3(c)
Give another expression of the form □×(□+□) equal to 14.
Steps only
2×(3+4) = 14
Answer
2×(3+4)
Question 3(d)
Give another expression of the form □×(□+□) equal to 14.
Steps only
7×(1+1) = 14
Answer
7×(1+1)
Question 4(i)
Find the sum of the numbers in grid 1 in two ways. Show at least two ways. Show at least two ways. Show at least two ways.
Steps only
5×4+4×8=52
(4+8+4)+(8+4+8)+(4+8+4)=16+20+16=52
| Column 1 | Column 2 | Column 3 |
|---|---|---|
| 4 | 8 | 4 |
| 8 | 4 | 8 |
| 4 | 8 | 4 |
Answer
The first grid totals 52. By values: 5×4+4×8=20+32=52. By rows: 16+20+16=52.
Question 4(ii)
Find the sum of the numbers in grid 2 in two ways. Show at least two ways. Show at least two ways. Show at least two ways.
Steps only
8×5+8×6=88
(5+6+6+5)×4=22×4=88
| Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|
| 5 | 6 | 6 | 5 |
| 6 | 5 | 5 | 6 |
| 6 | 5 | 5 | 6 |
| 5 | 6 | 6 | 5 |
Answer
The second grid totals 88. By values: 8×5+8×6=40+48=88. By rows: 22+22+22+22=88.
Figure it Out — Word problems and expressions
Textbook pages 42–44
Question 1(a)
Two orchards send 9 kg and 11 kg of mangoes daily. Find the weekly supply.
Steps only
7(9+11) = 7×20 = 140
Answer
140 kg.
Question 1(b)
Monthly income is ₹20000 and expenses are ₹5000, ₹5000 and ₹2000. Find the saving in 12 months.
Steps only
monthly saving = 20000−(5000+5000+2000) = 8000
annual saving = 12×8000 = 96000
Answer
₹96,000.
Question 1(c)
A snail climbs 3 cm each day and slips 2 cm each night on a 10 cm post. When does it reach the top?
Steps only
after each complete day-night pair: +1 cm
after 7 nights: 7 cm
day 8: 7+3 = 10 cm
Answer
Day 8.
Question 2
A reader finishes one two-page story daily except Tuesdays and Saturdays. Find the number of stories in 8 weeks and choose the matching expressions.
Steps only
reading days per week = 7−2 = 5
stories = (7−2)×8 = 40
7×8−2×8 = 40
Answer
40 stories; expressions (b) and (g).
Question 3(a)
Evaluate 1−2+3−4+5−6+7−8+9−10 in two ways.
Steps only
(1−2)+(3−4)+(5−6)+(7−8)+(9−10) = −5
(1+3+5+7+9)−(2+4+6+8+10) = 25−30 = −5
Answer
−5
Question 3(b)
Evaluate 1−1+1−1+1−1+1−1+1−1 in two ways.
Steps only
(1−1)+(1−1)+(1−1)+(1−1)+(1−1) = 0
5×1−5×1 = 0
Answer
0
Question 4(a)
Compare 49−7+8 and 49−7+8.
Steps only
49−7+8 = 49−7+8
Answer
=
Question 4(b)
Compare 83×42−18 and 83×40−18.
Steps only
83×42−18 > 83×40−18
Answer
>
Question 4(c)
Compare 145−17×8 and 145−17×6.
Steps only
145−17×8 < 145−17×6
Answer
<
Question 4(d)
Compare 23×48−35 and 23(48−35).
Steps only
23×48−35 > 23(48−35)
Answer
>
Question 4(e)
Compare (16−11)12 and −11×12+16×12.
Steps only
(16−11)12 = −11×12+16×12
Answer
=
Question 4(f)
Compare (76−53)88 and 88(53−76).
Steps only
(76−53)88 > 88(53−76)
Answer
>
Question 4(g)
Compare 25(42+16) and 25(43+15).
Steps only
25(42+16) = 25(43+15)
Answer
=
Question 4(h)
Compare 36(28−16) and 35(27−15).
Steps only
36(28−16) > 35(27−15)
Answer
>
Question 5(a)
Which listed expressions equal 83−37−12?
Steps only
84−38−12 = (83+1)−(37+1)−12 = 83−37−12
−37+83−12 = 83−37−12
Answer
Choices (i) and (iv).
Question 5(b)
Which listed expression equals 93+37×44+76?
Steps only
37×44+93+76 = 93+37×44+76
Answer
Choice (iv).
Question 6
Choose a number and make ten different expressions with that value.
Steps only
6+4 = 10
15−5 = 10
2×5 = 10
30÷3 = 10
7+8−5 = 10
3×4−2 = 10
(9−4)×2 = 10
18−(5+3) = 10
4×2+2 = 10
20−6−4 = 10
Answer
Ten examples with value 10 are shown above.
In-text explorations and Puzzle Time
Textbook pages 24–45
Question favourite-number
Choose 12 as a favourite number and write several expressions using two numbers and the four operations that have value 12.
Steps only
10+2=12
15−3=12
3×4=12
24÷2=12
Answer
Examples are 10+2, 15−3, 3×4 and 24÷2. There can be many more.
Question comparison-reasoning
Compare each pair using <, > or =: (a) 245+289 and 246+285; (b) 273−145 and 272−144; (c) 364+587 and 363+589; (d) 124+245 and 129+245; (e) 213−77 and 214−76.
Steps only
(a) first addend is 1 less but second is 4 more: first total is 3 more
(b) both numbers decrease by 1: values equal
(c) first addend is 1 more but second is 2 less: first total is 1 less
(d) second first addend is 5 greater
(e) second starts 1 more and subtracts 1 less: second result is 2 more
Answer
(a) >; (b) =; (c) <; (d) <; (e) <.
Question compare-marble-gains
Which is greater: 1023+125 or 1022+128? Explain without adding both totals from scratch.
Steps only
Joy starts 1 marble behind: 1022 is one less than 1023
Joy receives 3 more: 128 is three more than 125
Net change for Joy: −1+3=+2
Answer
1022+128 is greater by 2. Joy starts with one fewer marble but receives three more, so she finishes with two more.
Question compare-marble-losses
Compare 113−25 and 112−24. Which is greater?
Steps only
113 is 1 more than 112
25 is also 1 more than 24, so the first amount loses 1 extra
The changes cancel
Answer
They are equal: 113−25=112−24=88.
Question subtraction-inverse
Explain why subtracting a number gives the same result as adding its opposite. Check with 13−8.
Steps only
13−8=5
The opposite of 8 is −8
13+(−8)=5
Answer
Subtracting 8 and adding −8 both give 5, so 13−8=13+(−8). This works because a positive and its opposite cancel to zero.
Question reorder-terms
Does changing the order in which the terms of an addition are added change its value? Check using 18−7+4.
Steps only
18−7+4=18+(−7)+4
18+(−7)+4=15
4+18+(−7)=15
Answer
No. The value stays 15 because the terms are added; they may be rearranged.
Question terms-table
Complete the term list for: 13−2+6; 5+6×3; 4+15−9; 23−2×4+16; and 28+19−8.
Steps only
13−2+6=13+(−2)+6
5+6×3 has terms 5 and 6×3
4+15−9=4+15+(−9)
23−2×4+16=23+(−2×4)+16
28+19−8=28+19+(−8)
Answer
The term lists are: 13, −2, 6; 5, 6×3; 4, 15, −9; 23, −2×4, 16; and 28, 19, −8.
Question forgotten-addend
Manasa adds a list and gets 11,749, then notices she forgot the fourth number, 9,055. Must she start again? What is the corrected total?
Steps only
11,749+9,055=20,804
Answer
No. She can add the omitted 9,055 to her first total. The corrected sum is 20,804.
Question same-sum-token-model
The terms of (−7)+10+(−11) can be added in different orders. Show two orders and explain why the total stays the same.
Steps only
(−7)+10+(−11)=3−11=−8
(−7)+(−11)+10=−18+10=−8
Answer
Both orders give −8. In a token model, rearranging the same positive and negative tokens does not change how many unmatched tokens remain.
Question tinker-terms-table
Complete the chapter table by finding each value: 53+(−16), 54+(−16), 52+(−16), 53+(−15), 53+(−17), (−87)+(−16), (−88)+(−15), (−86)+(−18), and (−97)+(−26).
Steps only
53+(−16)=37
54+(−16)=38; 52+(−16)=36
53+(−15)=38; 53+(−17)=36
−87−16=−103; −88−15=−103; −86−18=−104; −97−26=−123
Answer
In order, the values are 37, 38, 36, 38, 36, −103, −103, −104 and −123.
Question seven-dosa-friends
Seven friends each buy a dosa for ₹23 and leave one ₹5 tip in total. Write an expression for the bill and identify its terms.
Steps only
Dosa cost: 7×23
Add the one tip: 7×23+5
Terms: 7×23 and 5
Answer
The expression is 7×23+5. Its terms are 7×23 and 5; its value is ₹166.
Question four-dosa-friends
Four friends each buy a dosa for ₹23 and leave one ₹5 tip. Write the expression and calculate the total.
Steps only
4×23+5=92+5=97
Answer
The total is ₹97, written 4×23+5. Its terms are 4×23 and 5.
Question two-shares-bill
Lhamo and Norbu each buy one cutlet for ₹43 and one rasgulla for ₹24. Write an expression for their total bill and evaluate it. Is 2×43+24 correct?
Steps only
Question grouping-five
There are 33 children playing in a grouping game. Write the groups-and-leftovers expression if the teacher calls 4, then if the teacher calls 7.
Steps only
For groups of 4: 33=8×4+1
For groups of 7: 33=4×7+5
Answer
For 4: 8 groups and 1 child left, so 8×4+1. For 7: 4 groups and 5 left, so 4×7+5.
Question class-size-grouping
If a class has n students and the teacher calls a group size k, how can you write the number in full groups and the students left over?
Steps only
Let q be the number of full groups and r the leftover count
n=q×k+r, where 0≤r<k
Answer
Write n=q×k+r, where q is the number of full groups and r is the number left over (0≤r<k). For example, with 33 students and groups of 5: 33=6×5+3.
Question rice-packets
Raghu already has four 2 kg rice packets. He buys another 100 kg and packs it into 2 kg packets. How many packets does he have altogether?
Steps only
New packets: 100÷2=50
Total: 4+100÷2=4+50
Answer
He has 54 packets. The expression is 4+100÷2, whose terms are 4 and 100÷2.
Question coin-combinations
Give two ways to make exactly ₹432 using ₹1 and ₹5 coins and ₹10, ₹20, ₹50 and ₹100 notes. Can you find another?
Steps only
4×100+1×20+1×10+2×1=432
8×50+1×10+4×5+2×1=432
4×100+3×10+2×1=432
Answer
Three ways are 4×100+1×20+1×10+2×1; 8×50+1×10+4×5+2×1; and 4×100+3×10+2×1. Each totals ₹432.
Question coin-expression-terms
For the two ways to pay ₹432, identify the terms in 4×100+1×20+1×10+2×1 and 8×50+1×10+4×5+2×1.
Steps only
First expression terms: 4×100, 1×20, 1×10, 2×1
Second expression terms: 8×50, 1×10, 4×5, 2×1
Answer
The terms are the separate amounts joined by plus signs: 4×100, 1×20, 1×10, 2×1; and 8×50, 1×10, 4×5, 2×1.
Question ruby-groups
Thirty-three students play a grouping game while Ruby sits out. If the teacher calls 5, explain the expression 6×5+3 and identify its terms.
Steps only
6×5=30 students in six full groups
33−30=3 students left
Terms: 6×5 and 3
Answer
Six groups contain 30 students, with 3 students left over. The expression is 6×5+3; its terms are 6×5 and 3.
Question bracket-marbles
Mallesh has 30 marbles. Arun has 5 bags with 4 marbles in each. Write and evaluate the total. Explain why 30+5×4 is not (30+5)×4.
Steps only
30+5×4=30+20=50
(30+5)×4=35×4=140
Answer
The total is 50 marbles, represented by 30+(5×4). The expression (30+5)×4 describes a different calculation and gives 140.
Question left-square-array
The left picture has 5 columns of 2 green squares and 3 red squares. Write an expression for the total and find the number of squares.
Steps only
5×2+3=10+3=13
Answer
There are 13 squares. The expression is 5×2+3: five columns of two green squares, plus three red squares.
Question right-square-array
In the right-hand arrangement, there are two columns, each with 5 yellow squares and 3 blue squares. Write and evaluate expressions for the total number of squares.
Steps only
2×(5+3)=2×8=16
5+3+5+3=16
2×5+2×3=10+6=16
Answer
There are 16 squares. Three equivalent expressions are 2×(5+3), 5+3+5+3 and 2×5+2×3.
Question change-money
Irfan buys items costing ₹15 and ₹56 and pays ₹100. Write an expression for his change and calculate it.
Steps only
100−(15+56)=100−71=29
Equivalently: 100−15−56=29
Answer
His change is ₹29.
Question hira-coins
Hira has 28 coins in one bag and 35 in another. She gives away 10 coins from the second bag. Write an expression for how many coins remain and evaluate it.
Steps only
28+(35−10)=28+25=53
28+35−10=53
Answer
Hira has 53 coins left. The expression 28+(35−10) can be written as 28+35−10 because the bracket is preceded by a plus sign.
Question bracket-distribution
Remove brackets from 500−(250−100), then evaluate. Why does the sign of 100 change?
Steps only
500−(250−100)=500−250+100=350
Answer
The expression equals 350. The subtraction applies to the whole bracket, so both signs inside change when the bracket is removed.
Question distributive-not-equal
Explain why 5×4+3 is not equal to 5×(4+3).
Steps only
5×4+3=20+3=23
5×(4+3)=5×7=35
Answer
They are not equal: the first is 23 and the second is 35.
Question commutative-distributive
Is 5×(4+3) equal to 5×(3+4) and to (3+4)×5?
Steps only
4+3=3+4=7
5×7=35
7×5=35
Answer
Yes. All three expressions equal 35.
Question parade-distribution
Four rows have 5 boy scouts each and three rows have 5 girl guides each. Write two expressions for the total and find it.
Steps only
4×5+3×5=20+15=35
(4+3)×5=7×5=35
Answer
There are 35 students. The expressions 4×5+3×5 and (4+3)×5 both give the total.
Question quick-products
Use the distributive property to find 95×8, 104×15 and 49×50.
Steps only
95×8=(100−5)×8=800−40=760
104×15=(100+4)×15=1,500+60=1,560
49×50=(50−1)×50=2,500−50=2,450
Answer
95×8=760; 104×15=1,560; 49×50=2,450.
Question known-product-adjustment
Given 53×18=954, use that fact to find 63×18.
Steps only
63×18=(53+10)×18
=53×18+10×18
=954+180=1,134
Answer
63×18=1,134.
Question ninety-seven-times-twenty-five
Use the distributive property to calculate 97×25.
Steps only
97×25=(100−3)×25
=100×25−3×25
=2,500−75=2,425
Answer
97×25=2,425.
Question other-quick-products
Give two products that are easier to calculate by using a nearby multiple of 10 or 100, then show the working.
Steps only
101×32=(100+1)×32=3,200+32=3,232
49×60=(50−1)×60=3,000−60=2,940
Answer
Examples: 101×32=3,232 and 49×60=2,940. Other suitable examples are possible.
Question story-eight-weeks
Melvin reads a two-page story each day except Tuesday and Saturday. How many stories does he complete in eight weeks, and how many pages does he read? Give equivalent expressions.
Steps only
He reads 5 days each week: 5×8=40 stories
Two pages per story: 40×2=80 pages
Also (7−2)×8×2=80 pages
Answer
He completes 40 stories and reads 80 pages. The expressions 5×2×8, (7−2)×2×8 and 7×2×8−2×2×8 each count the pages.
Question alternating-sum
Evaluate 1−2+3−4+5−6+7−8+9−10 in a convenient way.
Steps only
(1−2)+(3−4)+(5−6)+(7−8)+(9−10)
=−1−1−1−1−1
Answer
The value is −5.
Question equal-expression-identification
For 83−37−12, identify which are equal: (i) 84−38−12; (ii) 84−(37+12); (iii) 83−38−13; (iv) −37+83−12.
Steps only
Original: 83+(−37)+(−12)
(i) changes +1 and −1, so same
(ii) subtracts the same sum, so same
(iii) subtracts one extra in the last terms, so not same
(iv) rearranges the original terms, so same
Answer
Expressions
- (i) ,
- (ii) and
(iv) equal 83−37−12. Expression (iii) does not.
Question puzzle-time-1-20
Use four 4s to make each whole number from 1 to 20. For some targets, use concatenation (44), square root (√4=2) or factorial (4!=24), as well as +, −, ×, ÷ and brackets.
Steps only
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=(4!×√4+4)÷4; 14=4×4−4÷√4; 15=4×4−4÷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
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=(4!×√4+4)÷4; 14=4×4−4÷√4; 15=4×4−4÷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). Here 4!=24, √4=2, and 44 joins two 4s. Every expression uses exactly four 4 digits.
Question puzzle-time-number-set
Using 1, 2, 3, 4 and 5 exactly once in any order, use +, −, ×, ÷ and brackets to make as many whole-number values from −10 to 10 as possible.
Steps only
−10=1×(2−(3+(4+5)))
0=1×(2−(3+(4−5)))
10=1+(2+((3×4)−5))
Answer
Examples using each of 1, 2, 3, 4 and 5 exactly once: −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+(2÷(3+(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)).
Question puzzle-time-hundred
Use the digits 0 through 9 exactly once to form an expression with value 100. One valid answer is enough.
Steps only
123−45−67+89+0=100
Answer
One answer is 123−45−67+89+0=100. It uses each digit from 0 to 9 exactly once.