Instalments and hire purchase questions with answers

Questions

1. Question 1Total paid on hire purchase

   Devi buys a home theatre system on hire purchase. Devi pays a deposit of $288, followed by 24 monthly instalments of $125.28. Find the total amount Devi pays for it.

   Show the worked solution

   1. Step 1: The instalments: number of instalments × each instalment$24 \times 125.28 = 3006.72$
   2. Step 2: Total paid = deposit + all the instalments$\text{Total} = 288 + 3006.72$$\phantom{\text{Total}} = 3294.72$The total amount paid is $3294.72.
2. Question 2How much more than the cash price

   The table shows the terms for buying a washing machine on hire purchase. Find how much more the washing machine costs on hire purchase than for cash.

   Hire purchase	Washing machine
   Cash price	$960
   Deposit	10% of the cash price
   Monthly instalments	12 instalments of $74.88

   Show the worked solution

   1. Step 1: Deposit (in dollars)$\text{Deposit} = \dfrac{10}{100} \times 960$$\phantom{\text{Deposit}} = 96$
   2. Step 2: The instalments: number of instalments × each instalment$12 \times 74.88 = 898.56$
   3. Step 3: Hire-purchase price = deposit + all the instalments$96 + 898.56 = 994.56$
   4. Step 4: Extra paid = hire-purchase price − cash price$994.56 - 960 = 34.56$It costs $34.56 more on hire purchase.
3. Question 3The monthly instalment

   Nur buys a sofa set on the hire-purchase terms in the table. The balance after the deposit is charged simple interest for the 12 months, and the balance plus the interest is paid back in 12 equal monthly instalments. Find each monthly instalment.

   Hire purchase	Sofa set
   Cash price	$2,400
   Deposit	15% of the cash price
   Interest on the balance	5% per annum, simple interest
   Repayment	12 equal monthly instalments

   Show the worked solution

   1. Step 1: Deposit (in dollars)$\text{Deposit} = \dfrac{15}{100} \times 2400$$\phantom{\text{Deposit}} = 360$
   2. Step 2: The loan: cash price − deposit$\text{Loan} = 2400 - 360$$\phantom{\text{Loan}} = 2040$
   3. Step 3: Interest on the loan: I = PRT/100, with T in years$T = \dfrac{12}{12}$$\phantom{T} = 1$$I = \dfrac{2040 \times 5 \times 1}{100}$$\phantom{I} = 102$
   4. Step 4: Amount to pay back = loan + interest$2040 + 102 = 2142$
   5. Step 5: Each instalment = amount to pay back ÷ number of instalments$\dfrac{2142}{12} = 178.50$Each monthly instalment is $178.50.
4. Question 4Interest as a percentage of the loan

   Chen Wei buys a television on the hire-purchase terms in the table. The amount still owed after the deposit is the loan. Find the interest Chen Wei pays as a percentage of the loan.

   Hire purchase	Television
   Cash price	$3,600
   Deposit	10% of the cash price
   Monthly instalments	36 instalments of $106.20

   Show the worked solution

   1. Step 1: Deposit (in dollars)$\text{Deposit} = \dfrac{10}{100} \times 3600$$\phantom{\text{Deposit}} = 360$
   2. Step 2: The loan: cash price − deposit$\text{Loan} = 3600 - 360$$\phantom{\text{Loan}} = 3240$
   3. Step 3: The instalments: number of instalments × each instalment$36 \times 106.20 = 3823.20$
   4. Step 4: Interest = all the instalments − the loan$\text{Interest} = 3823.20 - 3240$$\phantom{\text{Interest}} = 583.20$
   5. Step 5: As a percentage of the loan$\dfrac{583.20}{3240} \times 100\% = 18\%$The interest is $18\%$ of the loan.
5. Question 5The rate of simple interest per annum

   Siti buys a home theatre system on the hire-purchase terms in the table. The amount still owed after the deposit is the loan, and the interest is simple interest on the loan for the 24 months. Find the rate of simple interest per annum.

   Hire purchase	Home theatre system
   Cash price	$4,200
   Deposit	10% of the cash price
   Monthly instalments	24 instalments of $182.70

   Show the worked solution

   1. Step 1: Deposit (in dollars)$\text{Deposit} = \dfrac{10}{100} \times 4200$$\phantom{\text{Deposit}} = 420$
   2. Step 2: The loan: cash price − deposit$\text{Loan} = 4200 - 420$$\phantom{\text{Loan}} = 3780$
   3. Step 3: The instalments: number of instalments × each instalment$24 \times 182.70 = 4384.80$
   4. Step 4: Interest = all the instalments − the loan$\text{Interest} = 4384.80 - 3780$$\phantom{\text{Interest}} = 604.80$
   5. Step 5: I = PRT/100, so R = 100I ÷ (P × T), with T in years$T = \dfrac{24}{12}$$\phantom{T} = 2$$R = \dfrac{100 \times 604.80}{3780 \times 2}$$\phantom{R} = 8$The rate of simple interest is $8\%$ per annum.
6. Question 6Work backwards to the cash price

   Jun Jie buys a piano on the hire-purchase terms in the table. The balance after the deposit is charged simple interest for the 24 months, and the balance plus the interest is paid back in the instalments shown. Find the cash price of the piano.

   Hire purchase	Piano
   Deposit	25% of the cash price
   Interest on the balance	6% per annum, simple interest
   Monthly instalments	24 instalments of $50.40

   Show the worked solution

   1. Step 1: All the instalments repay the loan plus its interest$24 \times 50.40 = 1209.60$
   2. Step 2: The interest is 6% a year for 2 years, so the loan grows to (100% + 6% × 2) of itself$100\% + 6 \times 2\% = 112\%$$\text{Loan} = \dfrac{100}{112} \times 1209.60$$\phantom{\text{Loan}} = 1080$
   3. Step 3: The loan is what is left after the 25% deposit: (100% − 25%) of the cash price$\text{Cash price} = \dfrac{100}{75} \times 1080$$\phantom{\text{Cash price}} = 1440$The cash price is $1440.
7. Question 7Buying a car: down payment and loan

   Ali buys a car on the terms in the table. Find the monthly instalment.

   Buying the car
   List price	$168,000
   Dealer's discount	8% of the list price
   Registration and other fees	$1,850 (paid in cash)
   Down payment	30% of the price after the discount
   Car loan	the rest of the price, at 2.5% per annum simple interest for 5 years, paid back in equal monthly instalments

   Show the worked solution

   1. Step 1: The price after the discount (in dollars)$\dfrac{92}{100} \times 168000 = 154560$
   2. Step 2: The down payment, and the loan (the rest)$\text{Down payment} = \dfrac{30}{100} \times 154560$$\phantom{\text{Down payment}} = 46368$$\text{Loan} = 154560 - 46368$$\phantom{\text{Loan}} = 108192$
   3. Step 3: Simple interest on the LOAN for the whole period: I = PRT/100$I = \dfrac{108192 \times 2.5 \times 5}{100}$$\phantom{I} = 13524$
   4. Step 4: Each instalment: (loan + interest) shared over the months$\dfrac{108192 + 13524}{60} = 2028.60$The monthly instalment is $2028.60.