Test 13: Algebraic Fractions II
Question 1: If you failed question 1 in the test, then you
must complete these 3 extra questions:
(a): Simplify: (i): a × bc (ii): 4x ÷ 2x
2b 3a 3 3
Clue: Part (i): First, cancel ANY 'top' with ANY 'bottom':
Cancelling the 'b's:
a × bc
2b 3a
And cancelling the 'a's:
┌———————————————————————— Since there is nothing left at the top of this
▼ fraction, we leave a 1 there...
1a × bc
2b 3a
Finally, multiplying the tops and multiplying the bottoms:
1 × c = c .
2 3 ...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
4x × 3 .
3 2x
Next, cancel ANY 'top' with ANY 'bottom':
Cancelling the 'x's:
4x × 3
3 2x
And cancelling the '3's:
┌—————————————————— Since there is nothing left at the top of this
▼ fraction, we leave a 1 there...
4x × 13 .
31 2x
And cancelling the '2's...
Finally, multiplying the tops and multiplying the bottoms...
(b): Simplify: (i): 3b × 8c (ii): 5x ÷ 3
4c 6 4 4
Clue: Part (i): First, cancel ANY 'top' with ANY 'bottom':
Cancelling the 'c's:
3b × 8c
4c 6
And cancelling down the '4' & '8':
3b × 28c
14c 6
And cancelling the '3's...
Finally, multiplying the tops and multiplying the bottoms...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
5x × 4.
4 3
Next, cancel ANY 'top' with ANY 'bottom':
Cancelling the '4's:
5x × 14.
41 3
Finally, multiplying the tops and multiplying the bottoms...
(c): Simplify: (i): 2c × d (ii): 1 ÷ 3 .
5d 4 6x 2xy
Clue: Sorry, no more help with this part....
Question 2: If you failed question 2 in the test, then you
must complete these 3 extra questions:
(a): Simplify: (i): a × b (ii): x ÷ y
3 2 4 3
Clue: Part (i): We can't cancel anything, so multiplying the tops and multiplying the bottoms:
a × b = ab .
3 2 ...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
x × 3 .
4 y
We can't cancel anything, so multiplying the tops and multiplying the bottoms...
(b): Simplify: (i): 5 × 2 (ii): 7 ÷ 1
bc c xy x
Clue: Sorry, there's no help for this part...
(c): Simplify: (i): 1 × 1 (ii): 3 ÷ 1
cd d a a
Clue: Sorry, you have to do this bit un-aided...
Question 3: If you failed question 3 in the test, then you
must complete these 3 extra questions:
(a): Simplify: (i): 2 × 3 . (ii): 2x ÷ 6 .
3a a-2 5 x-2
Clue: Part (i): To start with, it is important that we put in brackets (we can't cancel down a
fraction if there are and +/- sings sticking out of brackets...)
2 × 3
3a (a-2)
Cancelling the '3's:
2 × 31
13a (a-2)
Finally, multiplying the tops and multiplying the bottoms:
2 × 1 = ... .
a (a-2) a(a-2)
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
2x × (x-2).
5 6
Next, cancel ANY 'top' with ANY 'bottom':
Cancelling down the '2' & '6':
12x × (x-2)
5 63
Finally, multiplying the tops and multiplying the bottoms...
(b): Simplify: (i): b(b+1) × b+2 (ii): 2(x+3) ÷ 3(x+3)
b+2 b 5 10x
Clue: Part (i): Firstly, make sure all of the brackets are in place:
b(b+1) × (b+2)
(b+2) b
Then cancel down:
b(b+1) × (b+2)
(b+2) b1
And again:
(b+1) × (b+2)1
1(b+2) 1
Finally, multiplying the tops and multiplying the bottoms...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
2(x+3) × 10x.
5 3(x+3)
Next, cancel ANY 'top' with ANY 'bottom':
Cancelling the '(x+3)'s:
2(x+3) × 10x.
5 3(x+3)
Cancelling down the '10' & '5':
Finally, multiplying the tops and multiplying the bottoms...
(c): Simplify: (i): c × 6(c+2) (ii): x-1 ÷ 2(x-1)
3(c+2) 2 x 5
Clue: Sorry - you'll have to do this on your own...
Question 4: If you failed question 4 in the test, then you
must complete these 3 extra questions:
(a): Simplify: (i): a + 4 × 3a + 6 (ii): 3x ÷ 4
2a + 4 a + 4 3x - 9 2x - 6
Clue: Part (i): I know you are tempted to cancel it down like this:
a + 4 × 3a + 6
2a + 4 a + 4
But DON'T - IT IS WRONG!
The first step is always to ensure that there are BRACKETS around any sums in the
fractions - a fraction CANNOT be cancelled down if there are any +/- signs sticking out
of the brackets:
a + 4 × 3a + 6
2a + 4 a + 4
So, we have to put in brackets...
But, in this case, while we are putting in BRACKETS, we might as well factorise:
(a+4) × 3(a+2)
2(a+2) (a+4)
Now we can cancel down (ANY 'top' with ANY 'bottom')
1(a+4) × 3(a+2)
2(a+2) 1(a+4)
And again:
1(a+4) × 3(a+2)
2(a+2) 1(a+4)
Finally, multiplying the tops and multiplying the bottoms...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
3x × 2x - 6.
3x - 9 4
We can't cancel down until we put in the brackets and anyway, we need to FACTORISE:
3x × 2(x-3).
3(x-3) 4
Now cancel (ANY top with ANY bottom):
3x × 2(x-3).
3(x-3) 4
And again:
3x × 2(x-3).
3(x-3) 4
And again:
3x × 12(x-3).
3(x-3) 24
Finally, multiplying the tops and multiplying the bottoms...
(b): Simplify: (i): b × 2b + 10 (ii): x - 5 ÷ 2x - 10
b + 5 b² 4x 6x²
Clue: Part (i): The first step is always to ensure that there are BRACKETS around any sums in the
fractions (a fraction CANNOT be cancelled down if there are any +/- signs sticking out
of the brackets)...
But, in this case, while we are putting in BRACKETS, we might as well factorise:
b × 2(b+5)
(b+5) b²
Now we can cancel down (ANY 'top' with ANY 'bottom')
1b × 2(b+5)
(b+5) b²
And again:
Finally, multiplying the tops and multiplying the bottoms...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
x - 5 × 6x².
4x 2x - 10
We can't cancel down until we put in the brackets and anyway, we need to FACTORISE:
(x-5) × 6x².
4x 2(x-5)
Now cancel (ANY top with ANY bottom)...
And again...
And again...
Finally, multiplying the tops and multiplying the bottoms...
(c): Simplify: (i): 5c + 10 × 3c² (ii): 3x + 3 ÷ x + 1
2c c + 2 6 5
Clue: Sorry, no more help here - follow the method from part (a)
Question 5: If you failed question 5 in the test, then you
must complete these 3 extra questions:
(a): Simplify (i): a²+3a × 1 (ii): 1 ÷ 1
2 a+3 x+2 x²+3x+2
Clue: Part (i): The first step is always to ensure that there are BRACKETS around any sums in the
fractions (a fraction CANNOT be cancelled down if there are any +/- signs sticking out
of the brackets)...
But, in this case, while we are putting in BRACKETS, we might as well factorise:
a(a+3) × 1
2 (a+3)
Now we can cancel down (ANY 'top' with ANY 'bottom')
a(a+3) × 1
2 1(a+3)
Finally, multiplying the tops and multiplying the bottoms...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
1 × x²+3x+2.
x+2 1
We can't cancel down until we put in the brackets and anyway, we need to FACTORISE:
1 × (x+2)(x+1).
(x+2) 1
Now cancel (ANY top with ANY bottom):
1 × (x+2)(x+1).
1(x+2) 1
Finally, multiplying the tops and multiplying the bottoms...
(b): Simplify: (i): 2b²+2b × 9 (ii): 2 ÷ 4
3 2b+2 x²-25 x²-5x
Clue: Part (i): The first step is always to ensure that there are BRACKETS around any sums in the
fractions (a fraction CANNOT be cancelled down if there are any +/- signs sticking out
of the brackets)...
But, in this case, while we are putting in BRACKETS, we might as well factorise:
2b(b+1) × 9
3 2(b+1)
Now we can cancel down (ANY 'top' with ANY 'bottom')
2b(b+1) × 9
3 2(b+1)
And again...
2b(b+1) × 9
3 12(b+1)
And again (the '3' and the '9')...
Finally, multiplying the tops and multiplying the bottoms...
Part (ii): We can change the 'divide' to a 'multiply' if we turn the second fraction upside down:
1 × x²-5x.
x²-25 4
We can't cancel down until we put in the brackets and anyway, we need to FACTORISE:
1 × x(x-5).
(x+5)(x-5) 4
Now cancel (ANY top with ANY bottom)...
Finally, multiplying the tops and multiplying the bottoms...
(c): Simplify: (i): 4 × c-2 (ii): 3 ÷ 3
c²-2c 8 x²-4 2x-4
Clue: Sorry, I can't hold your hand for this question...
Note:
No answers are given for these extra revision questions.
If you are stuck, then make sure you have a serious
attempt the question and then stay behind at the end of
the lesson and ask me for help...
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