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📝 Investment mixture problems with interest (11 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 11 questions available

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What is Investment mixture problems with interest?

Definition:
Investment mixture problems with interest involve dividing a total amount of money into two or more investments with different interest rates, and the total interest earned is the sum of the interest from each investment, using the simple interest formula $I = Prt$ , and these problems are solved by setting up equations for the total principal and the total interest earned over a specific time period.

Working:
To solve these problems, create a table with rows for each investment, columns for principal ( $P$ ), interest rate ( $r$ ), time ( $t$ ), and interest ( $I$ ), then express the total interest as $I_1 + I_2 = \text{Total Interest}$ , and the total principal as $P_1 + P_2 = \text{Total Principal}$ , and solve the system, often assuming $t = 1$ year unless stated otherwise, so the equation simplifies to $P_1 r_1 + P_2 r_2 = \text{Total Interest}$ .

Example:
If $\5000 \) is invested in two accounts: one paying 4% and another paying 6%, and the total annual interest is $\260$ , let $x$ be invested at 4% and $5000 - x$ at 6%, then $0.04x + 0.06(5000 - x) = 260$ , solving gives $0.04x + 300 - 0.06x = 260$ , so $-0.02x = -40$ , hence $x = 2000$ invested at 4% and $3000$ invested at 6%.

Reason:
Investment mixture problems are crucial for financial planning, helping individuals and businesses allocate funds to maximize returns, manage risk, and achieve targeted income, and they also illustrate practical applications of algebra in personal finance.

Easy

Medium

Hard

📝 All Investment mixture problems with interest MCQs

Q1. An investor places $\4,000 \) at $5\%$ simple interest and the remainder at $8\%$ . If the total investment is $\10,000$ and the annual interest is $\$650$ , how much was invested at $8\%$ ?

\$3,000

\$4,000

\$5,000

✅

\$6,000

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Let $x$ be the amount invested at $8\%$ . Then $0.08x+0.05(10,000-x)=650$ . Simplifying gives $0.03x=150$ , so $x=5,000$ . Therefore, $\$5,000$ was invested at $8\%$ .

Q2. Which equation correctly models an investment of $\12,000 \), with $x$ dollars earning $6\%$ and the remaining amount earning $9\%$ , when the total annual interest is $\900$ ?

0.06x+0.09(12,000-x)=900

✅

0.06(12,000-x)+0.09x=900

0.06x+0.09(12,000+x)=900

0.15x(12,000)=900

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The amount earning $6\%$ is $x$ , so its interest is $0.06x$ . The remaining investment is $12,000-x$ , producing $0.09(12,000-x)$ . Adding these interest amounts gives the required total of 900.

Q3. An investor has $\20,000 \) to divide between two accounts. One earns $4\%$ and the other earns $7\%$ . If the desired annual interest is $\1,100$ , what percentage of the money must be placed in the $7\%$ account?

A.0.3

B.0.4

C.0.5 ✅

D.0.6

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: If $x$ dollars earn $7\%$ , then $20,000-x$ earns $4\%$ . The equation $0.07x+0.04(20,000-x)=1,100$ gives $0.03x=300$ , so $x=10,000$ . Thus exactly $50\%$ must earn $7\%$ .

Q4. A student says, 'Since the two interest rates are $5\%$ and $9\%$ , the average interest rate must be $7\%$ , so any mixture will earn $7\%$ .' What is the best evaluation of this reasoning?

A.It is always correct because averages ignore investment amounts.

B.It is correct only when equal amounts are invested at both rates. ✅

C.It is incorrect because simple interest cannot be averaged.

D.It is incorrect because the lower rate must always receive more money.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A simple average of $5\%$ and $9\%$ applies only when equal amounts are invested. If different amounts are invested, the overall rate is a weighted average, so the account receiving more money has greater influence on total interest.

Q5. A company invests $\15,000 \) between accounts earning $3\%$ and $6\%$ . The accountant claims that investing $\10,000$ at $3\%$ and $\5,000 \) at $6\%$ produces $\750$ interest. Is the claim correct?

A.Yes, because

3\%+6\%=9\%

B.Yes, because the average rate is

5\%

C.No, the actual interest is

\$600

. ✅

D.No, the actual interest is

\$900

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The first account earns $0.03(10,000)=300$ , while the second earns $0.06(5,000)=300$ . Their total is 600, not 750. The accountant incorrectly treated the rates as though the entire investment earned their average.

Q6. A graph shows total annual interest $I$ on a fixed $\$10,000$ investment as the amount $x$ placed at $8\%$ increases, while the remainder earns $5\%$ . Which feature should the graph have?

A.A horizontal line because total money is fixed.

B.A decreasing line because less money earns

5\%

C.An increasing line because each dollar shifted earns

3\%

more. ✅

D.A curved line because interest compounds over time.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Each dollar moved from the $5\%$ account to the $8\%$ account increases annual interest by $3\%$ of that dollar. Because the relationship is linear under simple interest, the graph rises steadily rather than curving.

Q7. A graph of annual interest versus dollars invested at $7\%$ starts at $\800 \) when $x=0$ and reaches $\1,400$ when $x=10,000$ . Assuming the remaining money earns a fixed lower rate, what does the slope represent?

A.The total amount invested

B.The lower interest rate only

C.The increase in annual interest per dollar moved to the higher-rate account ✅

D.The number of years in the investment

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The slope measures the change in total annual interest divided by the change in the amount shifted into the higher-rate account. Therefore, it represents the additional annual interest earned per dollar moved between the two rates.

Q8. A retiree invests $\25,000 \) between accounts earning $4\%$ and $7\%$ . She wants exactly $\1,450$ annual interest. If she later moves $\$2,000$ from the $4\%$ account to the $7\%$ account, what happens to annual interest?

A.It decreases by

\$60

B.It increases by

\$60

. ✅

C.It increases by

\$80

D.It remains unchanged.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Moving $\2,000 \) from $4\%$ to $7\%$ changes the annual interest by $0.07(2,000)-0.04(2,000)$ . This equals $140-80=60$ , so the annual interest increases by $\60$ .

Q9. An investor wants $\2,000 \) annual interest from $\30,000$ split between $5\%$ and $8\%$ accounts. A friend solves $0.05x+0.08x=2,000$ . What is the error?

A.The rates should be multiplied rather than added.

B.The second account should contain

30,000-x

, not

x

. ✅

C.The total interest should be divided by two.

D.The total investment should be

x+30,000

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Both accounts cannot contain the same variable $x$ because their amounts must add to $30,000$ . If $x$ is placed at $5\%$ , the other account contains $30,000-x$ , so the correct model is $0.05x+0.08(30,000-x)=2,000$ .

Q10. A fixed $\$50,000$ investment is divided between accounts earning $3\%$ and $9\%$ . What is the maximum possible annual interest if all restrictions allow any nonnegative division of the money?

\$1,500

\$3,000

\$4,500

✅

\$6,000

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: To maximize simple interest when the total investment is fixed and both choices are unrestricted, all money should be placed at the higher rate. Therefore $50,000(0.09)=4,500$ , which is greater than every mixture involving the $3\%$ account.

Q11. An investor wants an annual interest of $\1,800$ from $\30,000$ split between $4\%$ and $8\%$ accounts. What fraction of the money must earn $8\%$ ?

A.0.25

B.0.5 ✅

C.0.6

D.0.75

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The required average rate is $1,800/30,000=6\%$ . If a fraction $p$ earns $8\%$ , the weighted rate is $0.08p+0.04(1-p)=0.06$ . This gives $0.04p=0.02$ , so $p=0.5$ , or $50\%$ .

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