To prepare you for the rigors of Unit 1 General Mathematics, these warm-up questions focus on the financial modeling skills required for high-achieving students. We will move beyond simple calculations to explore the “Time Value of Money,” the impact of compounding periods, and the nuances of inflation.
Warm-up: Consumer Arithmetic (General Mathematics)
Q-1. An investor places $10,000 into a savings account that earns 4.5% per annum simple interest. How much interest will they have earned at the end of 6 years?
A. $3,022.60
B. $450
C. $12,700
D. $2,700
Hint: The formula for simple interest is I=Prt.
Q-2. A $5,000 investment earns 6% per annum interest compounded monthly. Which of the following correctly identifies the values for the periodic interest rate (r) and the number of periods (n) after 3 years?
A. r=0.005, n=3
B. r=0.06, n=36
C. r=0.06, n=3
D. r=0.005, n=36
Hint: Divide the annual rate by the number of compounds per year and multiply the years by the same number.
Q-3. If a pair of shoes costs $120 today and the average annual inflation rate is 3%, what is the predicted cost of the same shoes in 5 years, rounded to the nearest cent?
A. $139.11
B. $123.60
C. $156.00
D. $138.00
Hint: Think of inflation as a price that compounds annually like a bank account.
Q-4. The total price of a laptop, including GST, is . What was the price of the laptop before GST was added?
A. $1,485
B. $1,500
C. $1,815
D. $1,600
Hint: Remember that the final price is 110% of the original price.
Q-5. A stock price increases by in its first year but decreases by in its second year. How does the final price compare to the original price?
A. It is 2% lower than the original price.
B. It is 4% higher than the original price.
C. It is exactly the same as the original price.
D. It is 4% lower than the original price.
Hint: Try calculating the result starting with $100 as your base value.
Q-6. Which investment option provides the best return: a nominal rate of p.a. compounded annually, or a nominal rate of p.a. compounded monthly?
A. Neither, as both result in a loss after inflation.
B. The 12% compounded annually is better.
C. They are exactly the same.
D. The 11.8% compounded monthly is better.
Hint: Calculate the effective annual rate for the monthly compounding option to compare ‘apples to apples’.
Q-7. An antique car is valued at and is expected to appreciate at a rate of per year. Which formula models its value () after years?
A. V=50,000(0.08)t
B. V=50,000(1.08)t
C. V=50,000(0.92)t
D. V=50,000+(0.08×t)
Hint: Appreciation means the value is growing over time.
Q-8. A ‘Reducing Balance’ depreciation model for a piece of machinery worth uses a rate of p.a. What is its book value after years?
A. $14,450
B. $26,450
C. $14,000
D. $17,000
Hint: Reducing balance means the value is multiplied by (1−r) each year.
Q-9. If your bank account earns 5% interest but inflation is 3%, what is the ‘real’ rate of return on your investment?
A. 8%
B. 1.67%
C. 2%
D. 5%
Hint: Subtract the loss of purchasing power from the gain in interest.
Q-10. A student needs $10,000 in 4 years. If they can get a savings rate of 5% p.a. compounded annually, how much do they need to invest today? (Round to the nearest dollar)
A. $8,227
B. $8,000
C. $12,155
D. $7,835
Hint: You are looking for the ‘Principal’ (P) in the compound interest formula A=P(1+r)n.
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Great job completing the warm-up! These questions touched on some of the core themes of General Mathematics—specifically, how time and frequency of compounding can drastically change financial outcomes. Keep practicing these rearrangements of the interest formulas!
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ユニット1「一般数学」の難関に備えるために、これらのウォームアップ問題は、優秀な生徒に求められる財務モデリングスキルに焦点を当てています。単純な計算にとどまらず、「貨幣の時間価値」、複利期間の影響、そしてインフレのニュアンスまで掘り下げていきます。
問-1. ある投資家が、年利4.5%の単利付き普通預金口座に10,000ドルを預け入れました。6年後にはいくらの利息が得られるでしょうか？
ヒント：単利の公式はI=Prtです。
問-2. 5,000ドルを投資すると、月利6%の複利が得られます。3年後の期間利率（r）と期間数（n）の値を正しく示しているのは次のうちどれですか？
ヒント：年利率を年間複利回数で割り、同じ回数を年数に掛けます。
問-3. 靴1足の現在の価格が120ドルで、年間平均インフレ率が3%の場合、同じ靴の5年後の予測価格はいくらになるでしょうか？（端数は切り捨て）
ヒント：インフレは、銀行口座のように毎年複利で計算される価格と考えてください。
問-4. ノートパソコンの合計価格は、GST 10% を含めて 1,650 ドルです。GST が加算される前のノートパソコンの価格はいくらでしたか？
ヒント：最終価格は当初価格の 110% であることを覚えておいてください。
問-5. ある株価が 1 年目に 20% 上昇しましたが、2 年目に 20% 下落しました。最終価格は当初価格と比べてどうなりますか？
ヒント：基準価格を 100 ドルとして計算してみましょう。
問-6. 名目利率 12% を年複利で運用した場合と、名目利率 11.8% を月複利で運用した場合では、どちらの投資オプションの方がリターンが高いですか？
ヒント：同一条件で比較するために、月複利オプションの実効年利率を計算してください。
問-7.アンティークカーの価値はで、年間の割合で値上がりすると予想されます。$t$年後の価値（）をモデル化する式はどれですか？
ヒント：値上がりとは、価値が時間の経過とともに増加することを意味します。
問-8. 20,000$相当の機械の「逓減法」減価償却モデルでは、年率の割合で減価償却します。2年後の帳簿価額はいくらですか？
ヒント：逓減法とは、価値が毎年(1-r)倍になることを意味します。
問-9. 銀行口座の金利が5%で、インフレ率が3%の場合、投資の「実質」収益率はいくらですか？
ヒント：金利の上昇から購買力の低下を差し引きます。
問-10. ある学生が4年間で10,000ドル必要です。年利5%の貯蓄を毎年複利で得られる場合、今日投資する必要がある金額はいくらでしょうか？（端数は切り捨て）
ヒント：複利の式 A=P(1+r)n における「元本」（P）を求めます。
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Right answer
Q1. D. $2,700 Using $$, we calculate .
Q2. D. r=0.005, n=36 The monthly rate is and the total periods are .
Q3. A. $139.11 Inflation is calculated using the compound interest formula: .
Q4. B. $1,500 To find the pre-tax price, divide the total by : .
Q5. D. It is 4% lower than the original price. Multiplying the growth factors equals , which is a decrease.
Q6. D. The 11.8% compounded monthly is better.
The effective annual rate is , which beats .
Q7. B. Appreciation is modeled by a geometric growth formula where the base is plus the growth rate.
Q8. A. The value is calculated as .
Q9. C. 2% The ‘real’ rate is approximately the nominal rate minus the inflation rate ()
Q10. A. This is a Present Value calculation: .