Question increase-first-factor
By how much does 23×27 increase if 23 is increased by 1?
Steps only
(23+1)×27−23×27=27.
Answer
It increases by 27, the unchanged second factor.
CLASS 8 · GANITA PRAKASH PART I · CHAPTER 6
We Distribute, Yet Things Multiply
Use distributivity and algebraic identities to explain product changes, expand expressions, and solve number and area patterns.
105 worked answersAll 11 end-of-chapter questions
What you will learn
- Use distributivity to expand products and explain how products change.
- Apply square and difference-of-squares identities to calculate efficiently.
- Model tile, calendar and area patterns with algebra.
105 worked answers · short steps first, explanations when you need them
6.1 Some Properties of Multiplication
Textbook pages 137–141
Question increase-second-factor
By how much does 23×27 increase if 27 is increased by 1?
Steps only
23×(27+1)−23×27=23.
Answer
It increases by 23, the unchanged first factor.
Question increase-both-factors
By how much does 23×27 increase if both factors increase by 1?
Steps only
(a+1)(b+1)−ab=a+b+1.
For a=23,b=27: 23+27+1=51.
Answer
The product increases by 51.
Question increase-both-general
How much does ab increase when both a and b are increased by 1?
Steps only
(a+1)(b+1)=ab+a+b+1.
Subtract the original product ab.
Answer
The increase is a+b+1.
Question increase-and-decrease
Expand (a+1)(b−1).
Steps only
(a+1)(b−1)=ab−a+b−1.
Answer
The change from ab is b−a−1.
Question decrease-example
Give an example where decreasing both positive factors by 1 decreases their product.
Steps only
Choose a=10,b=3.
10×3=30, while (10−1)(3−1)=9×2=18.
Answer
The product decreases from 30 to 18.
Question change-by-m-n
If ab changes to (a+m)(b+n), how much does the product change?
Steps only
(a+m)(b+n)=ab+an+mb+mn.
Subtract ab.
Answer
The change is an+bm+mn.
Question change-minus2-plus3
Find the change in ab when the factors become (a−2) and (b+3).
Steps only
(a−2)(b+3)=ab+3a−2b−6.
Subtract ab.
Answer
The change is 3a−2b−6.
Question change-minus3-minus4
Find the change in ab when the factors become (a−3)(b−4).
Steps only
(a−3)(b−4)=ab−4a−3b+12.
Subtract ab.
Answer
The change is −4a−3b+12.
Question expand-increment-both-ways
Expand (a+1)(b+1) by first treating (b+1) as one term.
Steps only
(a+1)(b+1)=a(b+1)+1(b+1)=ab+a+b+1.
Answer
(a+1)(b+1)=ab+a+b+1. The increase from ab is a+b+1.
Question product-decreases-examples
Give three examples where increasing the first positive number by 1 and decreasing the second by 1 makes the product smaller.
Steps only
10×2=20; (10+1)(2−1)=11×1=11.
9×2=18; (9+1)(2−1)=10×1=10.
20×1=20; (20+1)(1−1)=21×0=0.
Answer
Examples: 20 becomes 11, 18 becomes 10, and 20 becomes 0; each product decreases.
Question product-change-negative-integers
For a=−5, b=8 and for a=−4, b=−5, compare ab with (a+1)(b−1).
Steps only
(−5)(8)=−40; (−5+1)(8−1)=−4×7=−28; change=12.
(−4)(−5)=20; (−4+1)(−5−1)=(−3)(−6)=18; change=−2.
Answer
The changes are +12 and −2, respectively, agreeing with b−a−1.
Question expand-two-negative-binomial-products
Expand (i) (a−u)(b+v) and (ii) (a−u)(b−v).
Steps only
(i) ab+av−ub−uv.
(ii) ab−av−ub+uv.
Answer
- (i) ab−ub+av−uv;
- (ii) ab−ub−av+uv.
Question verify-change-examples
Check the product changes when (i) a is reduced by 2 and b increased by 3, using a=8,b=6; (ii) a is reduced by 3 and b reduced by 4, using a=9,b=8.
Steps only
(i) 8×6=48; (8−2)(6+3)=6×9=54; change=+6.||Formula: 3a−2b−6=24−12−6=6.
(ii) 9×8=72; (9−3)(8−4)=6×4=24; change=−48.||Formula: −4a−3b+12=−36−24+12=−48.
Answer
The changes are +6 and −48, matching direct multiplication.
Expand and factor using distributivity
Textbook pages 142–144
Question pq-grid
If the centre of a 3×3 multiplication grid is pq, write all nine entries around it.
Steps only
The row factors are p−1,p,p+1.
The column factors are q−1,q,q+1.
Multiply the row factor by the column factor in each cell.
Answer
Rows: (p−1)(q−1), (p−1)q, (p−1)(q+1); p(q−1), pq, p(q+1); (p+1)(q−1), (p+1)q, (p+1)(q+1).
Question expand-3u-v3
Expand (3+u)(v−3).
Steps only
Multiply each term in the first bracket by every term in the second.
Collect like terms.
Answer
(3+u)(v−3)=uv+3v−3u−9.
Question expand-fraction
Expand (2/3)(15+6a).
Steps only
Multiply each term in the first bracket by every term in the second.
Collect like terms.
Answer
(2/3)(15+6a)=10+4a.
Question expand-10a-b
Expand (10a+b)(10c+d).
Steps only
Multiply each term in the first bracket by every term in the second.
Collect like terms.
Answer
(10a+b)(10c+d)=100ac+10ad+10bc+bd.
Question expand-3x-x6
Expand (3−x)(x−6).
Steps only
Multiply each term in the first bracket by every term in the second.
Collect like terms.
Answer
(3−x)(x−6)=−x²+9x−18.
Question expand-minus5
Expand (−5a+b)(c+d).
Steps only
Multiply each term in the first bracket by every term in the second.
Collect like terms.
Answer
(−5a+b)(c+d)=−5ac−5ad+bc+bd.
Question expand-5z-y9
Expand (5+z)(y+9).
Steps only
Multiply each term in the first bracket by every term in the second.
Collect like terms.
Answer
(5+z)(y+9)=5y+yz+9z+45.
Question unchanged-product-examples
Give three pairs (a,b) for which (a+2)(b−4)=ab.
Steps only
Expand: ab−4a+2b−8=ab.
So 2b=4a+8, hence b=2a+4.
Choose a=1,2,3.
Answer
Examples: (1,6), (2,8) and (3,10).
Question expand-polynomial-i
Expand (a+ab−3b²)(4+b).
Steps only
Multiply the polynomial by 4, then by b.
Combine the two ab terms.
Answer
4a+5ab+ab²−12b²−3b³.
Question expand-polynomial-ii
Expand (4y+7)(y+11z−3).
Steps only
4y(y+11z−3)=4y²+44yz−12y.
7(y+11z−3)=7y+77z−21.
Combine −12y+7y.
Answer
4y²+44yz+77z−5y−21.
Question difference-cubes
Expand (a−b)(a+b), (a−b)(a²+ab+b²) and (a−b)(a³+a²b+ab²+b³). What comes next?
Steps only
Expand and cancel opposite middle terms.
The results are a²−b², a³−b³ and a⁴−b⁴.
Answer
Next: (a−b)(a⁴+a³b+a²b²+ab³+b⁴)=a⁵−b⁵.
Question multiply-94-11
Calculate 94×11 using the place-value pattern.
Steps only
11=10+1, so 94×11=94×10+94.
Add the shifted copies with carries.
Answer
94×11=1034.
Question multiply-495-11
Calculate 495×11 using the place-value pattern.
Steps only
11=10+1, so 495×11=495×10+495.
Add the shifted copies with carries.
Answer
495×11=5445.
Question multiply-3279-11
Calculate 3279×11 using the place-value pattern.
Steps only
11=10+1, so 3279×11=3279×10+3279.
Add the shifted copies with carries.
Answer
3279×11=36069.
Question multiply-4791256-11
Calculate 4791256×11 using the place-value pattern.
Steps only
11=10+1, so 4791256×11=4791256×10+4791256.
Add the shifted copies with carries.
Answer
4791256×11=52703816.
Question multiply-by-101-rule
Describe a quick rule for dcba×101.
Steps only
101=100+1.
dcba×101=dcba00+dcba.
Answer
Add the number to itself shifted two places; align digits and carry as needed.
Question multiply-by-1001-rule
How can you multiply a number by 1001 or 10001 using place value?
Steps only
1001=1000+1, so append three zeroes and add the original number.
10001=10000+1, so append four zeroes and add the original number.
Answer
For 1001, add a three-place shift; for 10001, add a four-place shift.
Question by101-89
Evaluate 89×101 efficiently.
Steps only
101 is a power of ten plus 1; add one shifted copy.
Answer
89×101=8989.
Question by101-949
Evaluate 949×101 efficiently.
Steps only
101 is a power of ten plus 1; add one shifted copy.
Answer
949×101=95849.
Question by1001-265831
Evaluate 265831×1001 efficiently.
Steps only
1001 is a power of ten plus 1; add one shifted copy.
Answer
265831×1001=266096831.
Question by1001-1111
Evaluate 1111×1001 efficiently.
Steps only
1001 is a power of ten plus 1; add one shifted copy.
Answer
1111×1001=1112111.
Question by99-9734
Evaluate 9734×99 efficiently.
Steps only
99 is a power of ten minus 1; subtract one copy.
Answer
9734×99=963666.
Question by999-23478
Evaluate 23478×999 efficiently.
Steps only
999 is a power of ten minus 1; subtract one copy.
Answer
23478×999=23454522.
Question expand-fractional-binomial
Expand (3a/2)(a−b+1/5).
Steps only
(3a/2)(a−b+1/5)=(3a/2)a−(3a/2)b+(3a/2)(1/5).
=3a²/2−3ab/2+3a/10.
Answer
3a²/2−3ab/2+3a/10.
Question expand-sum-times-square-sum
Expand (a+b)(a²+2ab+b²) and combine like terms.
Steps only
a(a²+2ab+b²)+b(a²+2ab+b²).
=a³+2a²b+ab²+a²b+2ab²+b³.
=a³+3a²b+3ab²+b³.
Answer
a³+3a²b+3ab²+b³. The unlike terms in the expansion cannot be combined further.
Question multiply-3874-and-5678-by-11
Evaluate 3874×11 and 5678×11.
Steps only
3874×11=3874×(10+1)=38,740+3,874=42,614.
5678×11=5678×(10+1)=56,780+5,678=62,458.
Answer
3874×11=42,614 and 5678×11=62,458.
Question multiply-3874-by-101
Evaluate 3874×101 using place value.
Steps only
3874×101=3874×(100+1).
=387,400+3,874=391,274.
Answer
3874×101=391,274.
Question multiply-any-number-by-11
Describe a place-value method for multiplying a whole number of any number of digits by 11.
Steps only
N×11=N×10+N.
For 4,275: 4,275×11=47,025.
Answer
For every whole number N, N×11=N×10+N; add the one-place-shifted number to the original, carrying as needed.
Square identities and difference of squares
Textbook pages 145–150
Question square-binomial-a
What is the expansion of (a+b)²?
Steps only
(a+b)(a+b)=a²+ab+ab+b².
Answer
a²+2ab+b².
Question square-binomial-b
What is the expansion of (a−b)²?
Steps only
(a−b)(a−b)=a²−ab−ab+b².
Answer
a²−2ab+b².
Question difference-squares
Find (a+b)(a−b), and explain the difference-of-squares identity geometrically.
Steps only
Distribute: a²−ab+ab−b².
The middle terms cancel.
Answer
a²−b².
Question compare-opposite-squares
Which is greater: (a−b)² or (b−a)²?
Steps only
b−a=−(a−b).
Squaring removes the sign: (−x)²=x².
Answer
They are equal for every a and b.
Question 100-difference-squares
Give two square numbers whose difference is 100.
Steps only
676−576=100.
These are 26²−24².
Answer
100=26²−24².
Question square-406
Find 406² efficiently.
Steps only
Write 406 as a nearby round number plus or minus a small amount.
Apply (a±b)²=a²±2ab+b².
Answer
406²=164836.
Question square-72
Find 72² efficiently.
Steps only
Write 72 as a nearby round number plus or minus a small amount.
Apply (a±b)²=a²±2ab+b².
Answer
72²=5184.
Question square-145
Find 145² efficiently.
Steps only
Write 145 as a nearby round number plus or minus a small amount.
Apply (a±b)²=a²±2ab+b².
Answer
145²=21025.
Question square-1097
Find 1097² efficiently.
Steps only
Write 1097 as a nearby round number plus or minus a small amount.
Apply (a±b)²=a²±2ab+b².
Answer
1097²=1203409.
Question square-124
Find 124² efficiently.
Steps only
Write 124 as a nearby round number plus or minus a small amount.
Apply (a±b)²=a²±2ab+b².
Answer
124²=15376.
Question identities-integers-fractions
Do the distributive and square identities work for negative integers and fractions?
Steps only
The identities follow from multiplication and addition rules, not from a restriction to counting numbers.
Substitution of negative values or fractions preserves both sides.
Answer
Yes. They hold for negative integers and fractions as well.
Question square-area-65
A square has side 65 units. Split its side as 60+5 and find the four areas and the total area. Also check the splits 30+35 and 52+13.
Steps only
60²=3600; 60×5=300 (twice); 5²=25.||3600+300+300+25=4225.||30²+2(30×35)+35²=900+2100+1225=4225.||52²+2(52×13)+13²=2704+1352+169=4225.
Answer
The area is 4,225 square units in each split.
Question sum-square-compared-squares
For integers a and b, when is (a+b)² greater than a²+b²?
Steps only
(a+b)²−(a²+b²)=2ab.||Greater if ab>0; equal if ab=0; smaller if ab<0.
Answer
(a+b)² is greater exactly when ab>0.
Question square-sum-examples
Expand (m+3)² and (6+p)² using the identity; expand (6x+5)²; then expand (3j+2k)² using both the identity and the distributive property.
Steps only
(m+3)²=m²+6m+9; (6+p)²=p²+12p+36.
(6x+5)²=36x²+60x+25.
(3j+2k)²=9j²+12jk+4k².
Distribute: (3j+2k)(3j+2k)=9j²+6jk+6jk+4k².
Answer
m²+6m+9; p²+12p+36; 36x²+60x+25; 9j²+12jk+4k².
Question square-difference-examples
Expand (b−6)², (−2a+3)² and (7y−3z/4)² using both the square identity and distribution. Then find 99² and 58².
Steps only
(b−6)²=b²−12b+36; (−2a+3)²=4a²−12a+9.
(7y−3z/4)²=49y²−21yz/2+9z²/16.
99²=9801; 58²=3364.
Answer
The three expansions are b²−12b+36, 4a²−12a+9, and 49y²−21yz/2+9z²/16; 99²=9,801 and 58²=3,364.
Question square-pattern-sum-difference-identity
Show that twice the sum of two squares equals the square of their sum plus the square of their difference.
Steps only
(a+b)²+(a−b)²=(a²+2ab+b²)+(a²−2ab+b²)=2a²+2b².
Answer
2(a²+b²)=(a+b)²+(a−b)².
Question square-55-area-check
Use (60−5)² to find 55².
Steps only
55²=(60−5)²=60²−2×60×5+5².
=3600−600+25=3025.
Answer
55²=3,025.
Question square-104-and-37
Use a square identity to find 104² and 37².
Steps only
104²=(100+4)²=10000+800+16=10816.
37²=(40−3)²=1600−240+9=1369.
Answer
104²=10,816 and 37²=1,369.
Question products-98x102-45x55
Use a difference of squares to find 98×102 and 45×55.
Steps only
98×102=(100−2)(100+2)=10000−4=9996.
45×55=(50−5)(50+5)=2500−25=2475.
Answer
98×102=9,996 and 45×55=2,475.
Question square-nearby-31-and-197
Use a²=(a+b)(a−b)+b² to find 31² and 197².
Steps only
31²=32×30+1=961.
197²=200×194+9=38809.
Answer
31²=961 and 197²=38,809.
6.3 Mind the Mistake, Mend the Mistake
Textbook pages 150
Question 1
Check the simplification of −3p(−5p+2q). Decide whether it is correct; if not, write the correct result.
Steps only
−3p(−5p+2q)=15p²−6pq.
Answer
15p²−6pq.
Question 2
Check the simplification of ½(x−1)+3(x+4). Decide whether it is correct; if not, write the correct result.
Steps only
½(x−1)+3(x+4)=x/2−1/2+3x+12=7x/2+23/2.
Answer
7x/2+23/2.
Question 3
Check the simplification of y+2(y+2). Decide whether it is correct; if not, write the correct result.
Steps only
y+2(y+2)=y+2y+4=3y+4.
Answer
3y+4.
Question 4
Check the simplification of (5m+6n)². Decide whether it is correct; if not, write the correct result.
Steps only
(5m+6n)²=25m²+60mn+36n².
Answer
25m²+60mn+36n².
Question 5
Check the simplification of (−q+2)². Decide whether it is correct; if not, write the correct result.
Steps only
(−q+2)²=(−q)²+2(−q)(2)+2²=q²−4q+4.
Answer
Correct: q²−4q+4.
Question 6
Check the simplification of 3a(2b+3c). Decide whether it is correct; if not, write the correct result.
Steps only
3a(2b+3c)=(3a)(2b)+(3a)(3c)=6ab+9ac.
Answer
6ab+9ac.
Question 7
Check the simplification of ½(10s−6)+3. Decide whether it is correct; if not, write the correct result.
Steps only
½(10s−6)+3=5s−3+3=5s.
Answer
Correct: 5s.
Question 8
Check the simplification of 5w²+6w. Decide whether it is correct; if not, write the correct result.
Steps only
5w²+6w stays as 5w²+6w.
Answer
5w²+6w.
Question 9
Check the simplification of 2a³+3a³+6a²b+6ab². Decide whether it is correct; if not, write the correct result.
Steps only
2a³+3a³+6a²b+6ab²=5a³+6a²b+6ab².
Answer
5a³+6a²b+6ab².
Question 10
Check the simplification of (a+2)(b+4). Decide whether it is correct; if not, write the correct result.
Steps only
(a+2)(b+4)=ab+4a+2b+8.
Answer
ab+4a+2b+8.
Question 11
Check the simplification of ab²+a²b+a²b². Decide whether it is correct; if not, write the correct result.
Steps only
ab²+a²b+a²b²=ab(b+a+ab).
Answer
Correct: ab(a+b+ab).
Question 12
Check the simplification of (x+2)(x+5). Decide whether it is correct; if not, write the correct result.
Steps only
(x+2)(x+5)=x²+5x+2x+10=x²+7x+10.
Answer
Correct: x²+7x+10.
Patterns, areas and applications
Textbook pages 151–158
Question tile-pattern-count
A pattern has (n+2)²−n² tiles in Step n. Find the counts in Steps 1, 2, 3, 4 and 10.
Steps only
Simplify: (n+2)²−n²=4n+4.
Substitute n=1,2,3,4,10.
Answer
8, 12, 16, 20 and 44 tiles, respectively.
Question tile-pattern-general
If Step n is a square frame with outside side n+2 and inside side n, write its tile count and simplify.
Steps only
Count by area: (n+2)²−n².
Expand: n²+4n+4−n².
Answer
4n+4 tiles (equivalently (n+2)²−n²).
Question dashed-area
Write the area of the dashed region in two ways. Then substitute p=6, r=3.5 and s=9.
Steps only
(p−r)(s−r)=ps−pr−sr+r².
(6−3.5)(9−3.5)=2.5×5.5=13.75.
Answer
The area is 13.75 square units.
Question calendar-diagonal-products
For a 2×2 block of consecutive calendar dates, how do the diagonal products compare?
Steps only
Write the block as a, a+1 on the top row and a+7, a+8 below.
Diagonal products are a(a+8) and (a+1)(a+7).
Their difference is (a+1)(a+7)−a(a+8)=7.
Answer
The product on the diagonal with the two middle dates is 7 greater.
Question verify-k1
(k+1)(k+2)−(k+3) is always 2.
Steps only
Expand or factor the expression and simplify.
Answer
False. It simplifies to k²+2k−1, which depends on k.
Question verify-odd-product4
(2q+1)(2q−3) is a multiple of 4.
Steps only
Expand or factor the expression and simplify.
Answer
False. It equals 4q(q−1)−3, leaving remainder 1 modulo 4.
Question verify-odd-square8
An odd square is 1 more than a multiple of 8.
Steps only
Expand or factor the expression and simplify.
Answer
True. For odd n=2k+1, n²=4k(k+1)+1; k(k+1) is even, so 4k(k+1) is divisible by 8.
Question verify-square-minus5
(6n+2)²−(4n+3)² is 5 less than a square.
Steps only
Expand or factor the expression and simplify.
Answer
False. It simplifies to 20n²−5; this is not generally one less than a square.
Question remainders-product-7
One number leaves remainder 3 and another remainder 5 on division by 7. Find the remainders of sum, second minus first, and product.
Steps only
Sum: 3+5=8 leaves 1.
Difference: 5−3=2.
Product: 3×5=15 leaves 1.
Answer
Remainders: 1, 2 and 1.
Question middle-square-minus-neighbours
Choose n−1,n,n+1. What is n²−(n−1)(n+1)?
Steps only
(n−1)(n+1)=n²−1.
n²−(n²−1)=1.
Answer
The result is always 1.
Question half-square-sum
Write the result algebraically for numbers a and b.
Steps only
Their sum is a+b.
Half the sum is (a+b)/2.
Multiply: (a+b)×(a+b)/2.
Answer
(a+b)²/2, which is half the square of their sum.
Question compare-14x26
Which product is larger?
Steps only
14×26=(16−2)(24+2).
Expanding gives 16×24+32−48−4=16×24−20.
Answer
16×24 is larger by 20.
Question compare-25x75
Which product is larger?
Steps only
25×75=(26−1)(74+1).
Expanding gives 26×74+26−74−1=26×74−49.
Answer
26×74 is larger by 49.
Question park-path-area
A rectangular park surrounds two g×g square plots with a path of width w as shown. Find the area to be tiled.
Steps only
The outer rectangle is (2g+4w) by (g+2w).
Subtract the two green squares: (2g+4w)(g+2w)−2g².
Expand and simplify.
Answer
8w(w+g) square feet.
Question square-pattern-next
A pattern’s Step 1, 2 and 3 counts are (2+1)², (2+2)², (2+3)². Find Step 4 and Step 10.
Steps only
The count at Step k is (k+2)².
Step 4: 6²=36. Step 10: 12²=144.
9 small squares
16 small squares
25 small squares
36 small squares
Answer
36 units in Step 4 and 144 in Step 10.
Question circle-pattern-count
A circle pattern has k²+2k circles at Step k. Find the counts at Steps 4, 10 and 15, and give the next count after Step 3.
Steps only
Step 4: 4²+2×4=24.||Step 10: 10²+2×10=120.||Step 15: 15²+2×15=255.||Step 3 has 15 circles; Step 4 has 24.
Answer
Steps 4, 10 and 15 have 24, 120 and 255 circles.
Question interior-shaded-square-area
Four equal rectangles, each with area mn, surround a shaded square inside a larger square with side m+n. Find the shaded area in two ways and verify the expressions are equal.
Steps only
Outer square area minus rectangles: (m+n)²−4mn.||Inner square side n−m: (n−m)².||(m+n)²−4mn=m²+2mn+n²−4mn=m²−2mn+n²=(n−m)².
Answer
The shaded area is (m+n)²−4mn=(n−m)².
Question slanted-shaded-area-equivalent-methods
Three equal rectangles have dimensions x by y. The shaded region is described by x²−xy, x(x+2y)−3xy, and x(x−y). Show the expressions agree and evaluate them for x=8, y=3.
Steps only
x²−xy=x(x−y).||x(x+2y)−3xy=x²+2xy−3xy=x²−xy.||For x=8,y=3: 64−24=40.
Answer
All three methods give area x²−xy; for x=8 and y=3, the area is 40 square units.
Question coin-conjoin-triangle
A triangular arrangement has 10 coins, and the task is to turn it upside down by moving one coin at a time. How many moves are needed? What is the minimum for the next triangular arrangement of 15 coins? Give the general rule for T coins.
Steps only
10 coins: 3 moves.
15 coins: ⌊15/3⌋=5 moves.
For T coins in a triangular arrangement: ⌊T/3⌋ moves.
Answer
The 10-coin arrangement takes 3 moves; the 15-coin arrangement takes 5 moves. For a triangular arrangement with T coins, the rule is ⌊T/3⌋ moves.
6.4 Figure it Out: Questions 1–3
Textbook pages 154–155
Question Q1(i)
Find 46² using the square-of-a-sum identity.
Steps only
46²=(40+6)²=40²+2(40)(6)+6².
=1600+480+36=2116.
Answer
46²=2,116.
Question Q1(ii)
Find 397×403 using the difference-of-squares identity.
Steps only
397×403=(400−3)(400+3).
=400²−3²=160000−9=159991.
Answer
397×403=159,991.
Question Q1(iii)
Find 91² using the square-of-a-difference identity.
Steps only
91²=(100−9)²=100²−2(100)(9)+9².
=10000−1800+81=8281.
Answer
91²=8,281.
Question Q1(iv)
Find 43×45 using a difference of squares.
Steps only
43×45=(44−1)(44+1).
=44²−1²=1936−1=1935.
Answer
43×45=1,935.
Question Q2(i)
Expand (p−1)(p+11).
Steps only
p²+11p−p−11=p²+10p−11.
Answer
p²+10p−11.
Question Q2(ii)
Expand (3a−9b)(3a+9b).
Steps only
(3a)²−(9b)²=9a²−81b².
Answer
9a²−81b².
Question Q2(iii)
Expand −(2y+5)(3y+4).
Steps only
(2y+5)(3y+4)=6y²+8y+15y+20=6y²+23y+20.
−(2y+5)(3y+4)=−6y²−23y−20.
Answer
−6y²−23y−20.
Question Q2(iv)
Expand (6x+5y)².
Steps only
(6x)²+2(6x)(5y)+(5y)²=36x²+60xy+25y².
Answer
36x²+60xy+25y².
Question Q2(v)
Expand (2x−1/2)².
Steps only
(2x)²−2(2x)(1/2)+(1/2)².
=4x²−2x+1/4.
Answer
4x²−2x+1/4.
Question Q2(vi)
Expand (7p)(3r)(p+2).
Steps only
(7p)(3r)=21pr.
21pr(p+2)=21p²r+42pr.
Answer
21p²r+42pr.
Question Q3
Write expressions for (i) two more than a square number and (ii) the sum of the squares of two consecutive integers.
Steps only
(i) If the integer is s: s²+2.
(ii) If the consecutive integers are m and m+1: m²+(m+1)²=2m²+2m+1.
Answer
- (i) s²+2.
- (ii) m²+(m+1)², equivalently 2m²+2m+1.
Question Q11(b)
For the second basic-unit pattern, continue the sequence, find the count at Steps 4 and 10, and write a rule for Step y.
Steps only
Step 4: (4+1)²+4=29.
Step 10: (10+1)²+10=131.
Step y: (y+1)²+y.
Answer
Step 4 has 29 units; Step 10 has 131 units; Step y has (y+1)²+y units.