1. Simplify 28 – [26 – {2 + 5 × (6 – 3)}]
Answer:
28 – [26 – {2 + 5 × (6 – 3)}]
28 – [26 – {2 + 5 × 3}]
28 – [26 – {2 + 15}]
28 – [26 – 17]
28 – [26 – 17]
28 – 9
19
5
4
3
2. Simplify √32 − √32 + √125
Answer: Work out the terms to their simpler form with smaller values
3
5
4
√32 − √32 + √125
4
= √16 × 2 − 2 + 5
4
4
= √16 × √2 − 2 + 5
4
= 2 × √2 + 3
4
= 2 √2 + 3
4−3
3. Given −4, find the equivalent form with positive exponents.
𝑥
Answer:
𝑥4
43
4. Simplify 4(𝑥 𝑟 )2𝑝 × 3
Answer:
4(𝑥 𝑟 )2𝑝 × 3 = 4𝑥 2𝑝𝑟 × 3 = 12𝑥 2𝑝𝑟
5. The scientific notation of a number is 3.815 × 10−7 . What is the number?
Answer:
0.0000003815
Check the procedure for putting a number into scientific notation form
6. Find 𝑦 if log10 (4 − 2𝑦) = 2.
Answer:
log10 (4 − 2𝑦) = 2 implies 102 = 4 − 2𝑦
That is,
102 − 4 = −3𝑦
96 = −2𝑦
96
𝑦 = = −48
−2
7. Given log 5 = 𝑚 and log 2 = 𝑛, express log 50 in terms of 𝑚 and 𝑛.
Answer:
Express log 50 in terms of log 5 and log 2 (choose appropriate factors of 50 and
then apply rules of logarithms):
log 50 = log(5 × 5 × 2) = log 5 + log 5 + log 2 = 𝑚 + 𝑚 + 𝑛 = 2𝑚 + 𝑛
8. If log 5 = 𝑥 and log 4 = 𝑦, the value of 102𝑦+𝑥 is:
Answer:
log 5 = 𝑥 ⇒ 10𝑥 = 5
log 4 = 𝑦 ⇒ 10𝑦 = 4
102𝑦+𝑥 = 102𝑦 × 10𝑥 = (10𝑦 )2 × 10𝑥 = 42 × 5 = 80
1
9. A value is divided by a fraction . What is the percentage change?
8
Answer:
If value is 100, then we have
Percentage change:
800−100
100
100
1
8
= 800.
∗ 100 = 700%
Therefore it increased by 700%
10. A value is increased by 60% and then decreased by 40%. What is the percentage
change?
Answer:
If a value is 100, then increased by 60% means it is now 160
Now decrease the new value 160 by 40% to get another new value 96
Percentage change :
96−100
100
∗ 100 = −4%
Therefore it decreased by 4%
11. The reduced price of a product is R4660. A dealer would have gained 20% profit
with the original price of R4200. Is there any profit or loss? How much?
Answer: Find the cost price as follows:
SP = CP + %profit × CP
4200 = CP + 0.2 × CP
4200 = 1.2CP
CP =
4200
1.2
CP = 3500
Now SP − CP = 4660 − 3500 = 1160
Thus, there is a profit of R1160
12. The price of a product with VAT charge (at a rate of 20%) is R810. What is the
price without VAT?
Answer:
810
120%
= 675
13. Solve for 𝑥 in the equation 𝑥 − 2 + 4𝑥 = 5 + 2𝑥 is:
Answer:
Group like terms together on each side of the equation, i.e.,
𝑥 + 4𝑥 − 2𝑥 = 5 + 2
3𝑥 = 7
7
1
3
3
𝑥= =2
14. The sum of two numbers is 76, and one of them is 8 less than the other. Find the
bigger number.
Answer:
Solving word problems using algebra:
Let 𝑥 be the bigger number, then the smaller number is 𝑥 − 8,
Then the sum of the numbers is 𝑥 + (𝑥 − 8)
That is,
𝑥 + (𝑥 − 8) = 76
2𝑥 − 8 = 76
2𝑥 = 84
𝑥 = 46
Thus, the bigger number is 46
15. Marcus, Sally and Sammy decided to share 20 sweets. Marcus took 8 sweets and
Sally took three times as many as Sammy. How many sweets did Sammy
receive?
Answer:
Solving word problems using algebra
Let 𝑥 be the number of sweets received by Sammy, then Sally received 3𝑥
sweets.
Thus, the total number of sweets is 𝑥 + 3𝑥 + 8
Then
𝑥 + 3𝑥 + 8 = 20
4𝑥 + 8 = 20
4𝑥 = 12
𝑥=3
Sammy received 3 sweets.
Questions 16 and 17 are based on the following:
Consider the system of linear equations:
𝑥 − 3𝑦 = 8
5 − 𝑥 = 10 − 4𝑦
16. The solution for 𝑦 in this system is:
Answer:
Solving a system of two linear equations simultaneously using substitution
method.
𝑥 − 3𝑦 = 8 ……..(1)
5 − 𝑥 = 10 − 4𝑦 ……..(2)
From (1): 𝑥 = 3𝑦 + 8 …….(3)
Substitute (3) into (2):
5 − (3𝑦 + 8 ) = 10 − 4𝑦
5 − 3𝑦 − 8 = 10 − 4𝑦
−3𝑦 + 4𝑦 = 10 − 5 + 8
𝑦 = 13
17. Find the value of
𝑥+𝑦
2
.
Answer: First find the value of 𝑥
𝑥 = 3𝑦 + 8 = 3 × 13 + 8 = 47
𝑥+𝑦
2
=
47+13
2
= 30
18. An entertainment event attracts 220 attendees and draws in R11 000 in revenue
from ticket sales. A child’s ticket costs R20 and an adult’s ticket R60. Let 𝑥 be the
number of child attendees and 𝑦 the number of adult attendees.
Find the system of linear equations that best explain this scenario.
Answer:
Equation for the total number in attendance:
If there are 𝑥 children in attendance and 𝑦 adults in attendance, then the total
number in attendance is 𝑥 + 𝑦, i.e., 𝑥 + 𝑦 = 220
Equation for the total revenue:
If there are 𝑥 children in attendance and each child must pay R20, then the total
cost for children tickets is 20𝑥
If there are 𝑦 adults in attendance and each adult must pay R60, then the total
cost for children tickets is 60𝑦
Then the total revenue is 20𝑥 + 60𝑦, i.e., 20𝑥 + 60𝑦 = 11000
Therefore, the system of equations is:
𝑥 + 𝑦 = 220
20𝑥 + 60𝑦 = 11000
Questions 19 and 20 are based on the following:
At a particular store, three tablets and two cellphones of a certain brand cost a
total of R3500. It is also observed that two tablets and one cellphone cost a total
of R2600. Let 𝑥 be the price of a tablet and 𝑦 the price of a cellphone.
19. Find the system of equations that best explains this scenario is:
Answer:
3𝑥 + 2𝑦 = 3500
2𝑥 + 𝑦 = 2600
20. Find the price of one tablet device.
Answer:
Solve the system simultaneously for 𝑥 and 𝑦 as follows:
2𝑥 + 𝑦 = 2600 …….(1)
3𝑥 + 2𝑦 = 3500 …….(2)
From (1): 𝑦 = 2600 − 2𝑥 …….(3)
Substitute (3) into (1):
3𝑥 + 2(2600 − 2𝑥) = 3500
3𝑥 + 5200 − 4𝑥 = 3500
3𝑥 − 4𝑥 = 3500 − 5200
−𝑥 = −1700
𝑥 = 1700
Therefore one tablet device cost R1700.