SOLUTION: a coin bank contains $3.60 in half dollars and dimes. if there are 24 coins in all, how many of each type are there? what are the equations for this problem?

Question 596839: a coin bank contains $3.60 in half dollars and dimes. if there are 24 coins in all, how many of each type are there? what are the equations for this problem?
Answer by math-vortex(648) (Show Source): You can put this solution on YOUR website!
Hi there--
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[I] Define your variables
Let's have h be the number of half-dollars in the coin bank.
Let's have d be the number of dimes in the coin bank.
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The value of the half-dollars is $.50 times the number of half-dollars there are in the bank. One half-dollar is 0.50, two half-dollars are 2*(0.50)=$1.00, three are 3*(0.50)=$1.50. The algebraic expression is 0.50*h.
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Using similar reasoning, the value of the dimes in the bank is 0.10*d since one dime is $0.10.
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Set-up your equations:
First make an equation for the total number of coins. There are 24 coins in all, so
[the number of half-dollars] + [the number of dimes] = 24
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In algebra, we write,

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Now we will make an equation for the total value of the coins in the bank, $3.60.
[the value of the half-dollars] + [the value of the dimes] = $3.60
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In algebra, we write,

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[III] Solve the system of equations
I will use the substitution method. Write the first equation in terms of d.

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Substitute 24-h for d in the second equation.

Now clear the parentheses.

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Combine like terms.

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Subtract 2.40 from both sides of the equation to isolate the h-term and the constant term on opposite sides.

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Divide both sides of the equation by 0.40 to isolate h.

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The equation h=3 means that there are 3 half-dollars. Since there are 24 coins all together, there must be 21 dimes.
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We need to make sure these coins have a total value of $3.60. Three half-dollars is $1.50, and twenty-one dimes is $2.10. $1.50+2.10=$3.60. BINGO!
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That's it. Feel free to email me via gmail if any part of this explanation is unclear.
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Ms.Figgy
math.in.the.vortex@gmail.com

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