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Math  /  Numbers & Operations

QuestionQUESTION 4 Ayanda plans to travel to New Zealand. Her flight has been paid for by her parents. However, she will need spending money of about NZ\150.4.1Iftheexchangerateiscurrently8,8595RandtotheNZ$,howmuchwillsheneedinSouthAfricanRandinordertohaveenoughspendingmoney?4.2AyandahasjustreceivedagiftofR500fromhergrandmotherandhergrandmotherpromisestogiveheranotherR500inayear′stime.Ayandadecidestoinvestthismoneyinanaccountwhichpays 150. 4.1 If the exchange rate is currently 8,8595 Rand to the NZ\$, how much will she need in South African Rand in order to have enough spending money? 4.2 Ayanda has just received a gift of R500 from her grandmother and her grandmother promises to give her another R500 in a year's time. Ayanda decides to invest this money in an account which pays 8,5 \%$ p.a. compounded annually. How much will she have in her account after two years? 4.3 How much additional money will she have to add to her investment after two years if she withdraws all the money for her trip? โ101 QUESTION 5 5.1 A bag contains 7 red marbles and 5 green marbles. One marble is drawn out of the bag at random.
Calculate the probability that it is: 5.1.1 a red marble 5.1.2 an orange marble 5.1.3 a red or a green marble 5.1.4 not a red marble

Studdy Solution

STEP 1

1. For problem 4.1, the exchange rate from South African Rand (ZAR) to New Zealand Dollar (NZD) is given.
2. For problem 4.2, the interest rate is compounded annually.
3. For problem 4.3, Ayanda will withdraw all the money after two years.
4. For problem 5.1, the total number of marbles is known, and the probabilities are calculated based on simple ratios.

STEP 2

1. Calculate the amount of money Ayanda needs in South African Rand for her spending money.
2. Determine how much Ayanda will have in her account after two years, given the interest rate.
3. Calculate the additional money Ayanda needs to add after two years if she withdraws all the money for her trip.
4. Calculate the probabilities of drawing different colored marbles from the bag.

STEP 3

Calculate the amount of money Ayanda needs in South African Rand for her spending money. Given: Exchange rate=8.8595 ZAR per NZD \text{Exchange rate} = 8.8595 \, \text{ZAR per NZD} Spending money required=150 NZD \text{Spending money required} = 150 \, \text{NZD}
Amount in ZAR: 150 NZD×8.8595 ZAR/NZD=1328.925 ZAR 150 \, \text{NZD} \times 8.8595 \, \text{ZAR/NZD} = 1328.925 \, \text{ZAR}

STEP 4

Determine how much Ayanda will have in her account after two years, given the interest rate. Initial investment: P=500 ZAR P = 500 \, \text{ZAR}
Interest rate: r=8.5% per annum r = 8.5\% \text{ per annum}
Total time: t=2 years t = 2 \text{ years}
Future value formula for compound interest: A=P(1+r)t A = P(1 + r)^t
First year: A1=500×(1+0.085)=500×1.085=542.5 ZAR A_1 = 500 \times (1 + 0.085) = 500 \times 1.085 = 542.5 \, \text{ZAR}
Adding the second R500 gift: P=542.5+500=1042.5 ZAR P = 542.5 + 500 = 1042.5 \, \text{ZAR}
Second year: A2=1042.5×(1+0.085)=1042.5×1.085=1131.7125 ZAR A_2 = 1042.5 \times (1 + 0.085) = 1042.5 \times 1.085 = 1131.7125 \, \text{ZAR}
Rounded to two decimal places: A2=1131.71 ZAR A_2 = 1131.71 \, \text{ZAR}

STEP 5

Calculate the additional money Ayanda needs to add after two years if she withdraws all the money for her trip. Total amount after two years: A2=1131.71 ZAR A_2 = 1131.71 \, \text{ZAR}
Amount needed: M=1328.925 ZAR M = 1328.925 \, \text{ZAR}
Additional money required: Additional money=1328.925−1131.71=197.215 ZAR \text{Additional money} = 1328.925 - 1131.71 = 197.215 \, \text{ZAR}
Rounded to two decimal places: Additional money=197.22 ZAR \text{Additional money} = 197.22 \, \text{ZAR}

STEP 6

Calculate the probabilities of drawing different colored marbles from the bag. Total number of marbles: Total marbles=7 red+5 green=12 marbles \text{Total marbles} = 7 \, \text{red} + 5 \, \text{green} = 12 \, \text{marbles}

STEP 7

Calculate the probability of drawing a red marble. P(red)=712 P(\text{red}) = \frac{7}{12}

STEP 8

Calculate the probability of drawing an orange marble. Since there are no orange marbles: P(orange)=0 P(\text{orange}) = 0

STEP 9

Calculate the probability of drawing a red or green marble. P(red or green)=P(red)+P(green) P(\text{red or green}) = P(\text{red}) + P(\text{green}) P(red or green)=712+512=1212=1 P(\text{red or green}) = \frac{7}{12} + \frac{5}{12} = \frac{12}{12} = 1

STEP 10

Calculate the probability of not drawing a red marble. P(not red)=1−P(red) P(\text{not red}) = 1 - P(\text{red}) P(not red)=1−712=512 P(\text{not red}) = 1 - \frac{7}{12} = \frac{5}{12}
Solution Summary:
1. Ayanda needs 1328.93 ZAR for spending money.
2. Ayanda will have 1131.71 ZAR in her account after two years.
3. Ayanda needs to add 197.22 ZAR to her investment after two years.
4. Probability calculations: - Red marble: 712\frac{7}{12} - Orange marble: 00 - Red or green marble: 11 - Not a red marble: 512\frac{5}{12}

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