Math / Algebra

QuestionJohn invested his inheritance of \ $12,000 in three different funds: part in a money-market fund paying 3\% interest annually; part in municipal bonds paying$ 4 \% $annually; and the rest in mutual funds paying$ 7 \% $annually. John invested$ \ $4,000$ more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $\$ 670$ . How much did he invest in each type of fund? a. Money Market-1,000
Municipal Bond-4,000 Mutual Funds-8,000 b. Money Market-2,000
Municipal Bond-3,000 Mutual Funds-7,000 c. Money Market-3,000
Municipal Bond-2,000 Mutual Funds-7,000 d. Money Market-7,000
Municipal Bond-2,000 Mutual Funds-1,000

Studdy Solution

STEP 1

What is this asking? How can we split $\$12,000$ into three funds with different interest rates so that we earn $\$670$ in interest, knowing that $\$4,000$ more was put in mutual funds than municipal bonds? Watch out! Don't mix up the interest rates or the investment amounts!
Also, make sure all the money is accounted for!

STEP 2

1. Set up the variables
2. Build the equations
3. Solve the system
4. Calculate the investments

STEP 3

Let's use variables to represent the unknowns.
Let $m$ be the amount invested in the money-market fund, $b$ be the amount invested in municipal bonds, and $f$ be the amount invested in mutual funds.
This makes it easy to keep track of everything!

STEP 4

The total investment is $\$12,000$ , so we have our first equation: $m + b + f = 12000$

STEP 5

John invested $\$4,000$ more in mutual funds than municipal bonds.
This gives us our second equation: $f = b + 4000$

STEP 6

The total interest earned is $\$670$ .
Remember to convert percentages to decimals by dividing by one hundred for calculations!
This gives us our third equation: $0.03m + 0.04b + 0.07f = 670$

STEP 7

Substitute the second equation ( $f = b + 4000$ ) into the first and third equations to eliminate $f$ .
This is like a strategic swap!
Our new equations are: $m + b + (b + 4000) = 12000 \implies m + 2b = 8000$ $0.03m + 0.04b + 0.07(b + 4000) = 670 \implies 0.03m + 0.11b = 390$

STEP 8

Now, let's solve for $m$ in terms of $b$ using the simplified first equation: $m = 8000 - 2b$

STEP 9

Substitute this expression for $m$ into the simplified third equation: $0.03(8000 - 2b) + 0.11b = 390$ $240 - 0.06b + 0.11b = 390$ $0.05b = 150$ $b = \frac{150}{0.05} = 3000$ So, John invested $\$3,000$ in municipal bonds.

STEP 10

Now, substitute $b = 3000$ back into the equation $m = 8000 - 2b$ : $m = 8000 - 2(3000) = 2000$ John invested $\$2,000$ in the money-market fund.

STEP 11

Finally, substitute $b = 3000$ into the equation $f = b + 4000$ : $f = 3000 + 4000 = 7000$ John invested $\$7,000$ in mutual funds.

STEP 12

We found that $m = 2000$ , $b = 3000$ , and $f = 7000$ .
Let's verify our solution: $2000 + 3000 + 7000 = 12000$ $7000 = 3000 + 4000$ $0.03(2000) + 0.04(3000) + 0.07(7000) = 60 + 120 + 490 = 670$ Everything checks out!

STEP 13

John invested $\$2,000$ in the money-market fund, $\$3,000$ in municipal bonds, and $\$7,000$ in mutual funds.
This corresponds to option (b).

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Studdy Solution

STEP 1

STEP 2

1. Set up the variables
2. Build the equations
3. Solve the system
4. Calculate the investments

STEP 3

Let's use variables to represent the unknowns.
Let $m$ be the amount invested in the money-market fund, $b$ be the amount invested in municipal bonds, and $f$ be the amount invested in mutual funds.
This makes it easy to keep track of everything!

STEP 4

The total investment is $\$12,000$ , so we have our first equation: $m + b + f = 12000$

STEP 5

John invested $\$4,000$ more in mutual funds than municipal bonds.
This gives us our second equation: $f = b + 4000$

STEP 6

The total interest earned is $\$670$ .
Remember to convert percentages to decimals by dividing by one hundred for calculations!
This gives us our third equation: $0.03m + 0.04b + 0.07f = 670$

STEP 7

Substitute the second equation ( $f = b + 4000$ ) into the first and third equations to eliminate $f$ .
This is like a strategic swap!
Our new equations are: $m + b + (b + 4000) = 12000 \implies m + 2b = 8000$ $0.03m + 0.04b + 0.07(b + 4000) = 670 \implies 0.03m + 0.11b = 390$

STEP 8

Now, let's solve for $m$ in terms of $b$ using the simplified first equation: $m = 8000 - 2b$

STEP 9

Substitute this expression for $m$ into the simplified third equation: $0.03(8000 - 2b) + 0.11b = 390$ $240 - 0.06b + 0.11b = 390$ $0.05b = 150$ $b = \frac{150}{0.05} = 3000$ So, John invested $\$3,000$ in municipal bonds.

STEP 10

Now, substitute $b = 3000$ back into the equation $m = 8000 - 2b$ : $m = 8000 - 2(3000) = 2000$ John invested $\$2,000$ in the money-market fund.

STEP 11

Finally, substitute $b = 3000$ into the equation $f = b + 4000$ : $f = 3000 + 4000 = 7000$ John invested $\$7,000$ in mutual funds.

STEP 12

We found that $m = 2000$ , $b = 3000$ , and $f = 7000$ .
Let's verify our solution: $2000 + 3000 + 7000 = 12000$ $7000 = 3000 + 4000$ $0.03(2000) + 0.04(3000) + 0.07(7000) = 60 + 120 + 490 = 670$ Everything checks out!

STEP 13

John invested $\$2,000$ in the money-market fund, $\$3,000$ in municipal bonds, and $\$7,000$ in mutual funds.
This corresponds to option (b).

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Studdy Solution

STEP 1

STEP 2

1. Set up the variables2. Build the equations3. Solve the system4. Calculate the investments

STEP 3

Let's use variables to represent the unknowns.Let mmm be the amount invested in the **money-market fund**, bbb be the amount invested in **municipal bonds**, and fff be the amount invested in **mutual funds**.This makes it easy to keep track of everything!

STEP 4

The total investment is $12,000\$12,000$12,000, so we have our first equation: m+b+f=12000m + b + f = 12000m+b+f=12000

STEP 5

John invested $4,000\$4,000$4,000 more in mutual funds than municipal bonds.This gives us our second equation: f=b+4000f = b + 4000f=b+4000

STEP 6

The total interest earned is $670\$670$670.Remember to convert percentages to decimals by dividing by one hundred for calculations!This gives us our third equation: 0.03m+0.04b+0.07f=6700.03m + 0.04b + 0.07f = 6700.03m+0.04b+0.07f=670

STEP 7

STEP 8

Now, let's solve for mmm in terms of bbb using the simplified first equation: m=8000−2bm = 8000 - 2bm=8000−2b

STEP 9

STEP 10

Now, **substitute** b=3000b = 3000b=3000 back into the equation m=8000−2bm = 8000 - 2bm=8000−2b: m=8000−2(3000)=2000m = 8000 - 2(3000) = 2000m=8000−2(3000)=2000 John invested $2,000\$2,000$2,000 in the money-market fund.

STEP 11

Finally, **substitute** b=3000b = 3000b=3000 into the equation f=b+4000f = b + 4000f=b+4000: f=3000+4000=7000f = 3000 + 4000 = 7000f=3000+4000=7000 John invested $7,000\$7,000$7,000 in mutual funds.

STEP 12

STEP 13

John invested $2,000\$2,000$2,000 in the money-market fund, $3,000\$3,000$3,000 in municipal bonds, and $7,000\$7,000$7,000 in mutual funds.This corresponds to option (b).

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1. Set up the variables
2. Build the equations
3. Solve the system
4. Calculate the investments

Let's use variables to represent the unknowns.
Let $m$ be the amount invested in the money-market fund, $b$ be the amount invested in municipal bonds, and $f$ be the amount invested in mutual funds.
This makes it easy to keep track of everything!

The total investment is $\$12,000$ , so we have our first equation: $m + b + f = 12000$

John invested $\$4,000$ more in mutual funds than municipal bonds.
This gives us our second equation: $f = b + 4000$

The total interest earned is $\$670$ .
Remember to convert percentages to decimals by dividing by one hundred for calculations!
This gives us our third equation: $0.03m + 0.04b + 0.07f = 670$

Now, let's solve for $m$ in terms of $b$ using the simplified first equation: $m = 8000 - 2b$

Now, substitute $b = 3000$ back into the equation $m = 8000 - 2b$ : $m = 8000 - 2(3000) = 2000$ John invested $\$2,000$ in the money-market fund.

Finally, substitute $b = 3000$ into the equation $f = b + 4000$ : $f = 3000 + 4000 = 7000$ John invested $\$7,000$ in mutual funds.

John invested $\$2,000$ in the money-market fund, $\$3,000$ in municipal bonds, and $\$7,000$ in mutual funds.
This corresponds to option (b).