Solved on Jan 25, 2024

Find the number of nickels and quarters that sum to $6.25 with a total of 37 coins.

STEP 1

Assumptions
1. The total amount of money is $6.25. 2. The money is made up of only nickels and quarters. 3. There are 37 coins in total. 4. Let$ x $represent the number of nickels. 5. Let$ y $represent the number of quarters. 6. The value of a nickel is$ 0.05 and the value of a quarter is $0.25.

STEP 2

First, we will set up the equation that represents the total number of coins.
$x + y = 37$
This equation states that the sum of the number of nickels ( $x$ ) and the number of quarters ( $y$ ) is equal to the total number of coins, which is 37.

STEP 3

Next, we will set up the equation that represents the total value of the coins in dollars.
$0.05x + 0.25y = 6.25$
This equation states that the value of the nickels (5 cents each) plus the value of the quarters (25 cents each) equals $6.25.

STEP 4

Now we have a system of two equations with two variables:
\begin{align} x + y &= 37 \\ 0.05x + 0.25y &= 6.25 \end{align}

STEP 5

We can express the system in matrix form as:
$\begin{bmatrix} 1 & 1 \\ 0.05 & 0.25 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 37 \\ 6.25 \end{bmatrix}$

STEP 6

To solve the system, we can use either substitution or elimination. For this problem, let's use substitution. We will solve the first equation for $x$ :
$x = 37 - y$

STEP 7

Now we will substitute $x = 37 - y$ into the second equation:
$0.05(37 - y) + 0.25y = 6.25$

STEP 8

Distribute $0.05$ to both terms inside the parentheses:
$1.85 - 0.05y + 0.25y = 6.25$

STEP 9

Combine like terms:
$1.85 + 0.20y = 6.25$

STEP 10

Subtract $1.85$ from both sides to isolate the term with $y$ :
$0.20y = 6.25 - 1.85$

STEP 11

Calculate the right side of the equation:
$0.20y = 4.40$

STEP 12

Divide both sides by $0.20$ to solve for $y$ :
$y = \frac{4.40}{0.20}$

STEP 13

Calculate the value of $y$ :
$y = 22$

STEP 14

Now that we have the value of $y$ , we can substitute it back into the equation $x = 37 - y$ to find the value of $x$ :
$x = 37 - 22$

STEP 15

Calculate the value of $x$ :
$x = 15$
So, we have $x = 15$ and $y = 22$ . This means there are 15 nickels and 22 quarters.
$x:$ 15 nickels $y:$ 22 quarters

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Find the number of nickels and quarters that sum to $6.25 with a total of 37 coins.

STEP 1

STEP 2

First, we will set up the equation that represents the total number of coins.
$x + y = 37$
This equation states that the sum of the number of nickels ( $x$ ) and the number of quarters ( $y$ ) is equal to the total number of coins, which is 37.

STEP 3

Next, we will set up the equation that represents the total value of the coins in dollars.
$0.05x + 0.25y = 6.25$
This equation states that the value of the nickels (5 cents each) plus the value of the quarters (25 cents each) equals $6.25.

STEP 4

Now we have a system of two equations with two variables:
\begin{align} x + y &= 37 \\ 0.05x + 0.25y &= 6.25 \end{align}

STEP 5

We can express the system in matrix form as:
$\begin{bmatrix} 1 & 1 \\ 0.05 & 0.25 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 37 \\ 6.25 \end{bmatrix}$

STEP 6

To solve the system, we can use either substitution or elimination. For this problem, let's use substitution. We will solve the first equation for $x$ :
$x = 37 - y$

STEP 7

Now we will substitute $x = 37 - y$ into the second equation:
$0.05(37 - y) + 0.25y = 6.25$

STEP 8

Distribute $0.05$ to both terms inside the parentheses:
$1.85 - 0.05y + 0.25y = 6.25$

STEP 9

Combine like terms:
$1.85 + 0.20y = 6.25$

STEP 10

Subtract $1.85$ from both sides to isolate the term with $y$ :
$0.20y = 6.25 - 1.85$

STEP 11

Calculate the right side of the equation:
$0.20y = 4.40$

STEP 12

Divide both sides by $0.20$ to solve for $y$ :
$y = \frac{4.40}{0.20}$

STEP 13

Calculate the value of $y$ :
$y = 22$

STEP 14

Now that we have the value of $y$ , we can substitute it back into the equation $x = 37 - y$ to find the value of $x$ :
$x = 37 - 22$

STEP 15

Calculate the value of $x$ :
$x = 15$
So, we have $x = 15$ and $y = 22$ . This means there are 15 nickels and 22 quarters.
$x:$ 15 nickels $y:$ 22 quarters

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Find the number of nickels and quarters that sum to $6.25 with a total of 37 coins.

STEP 1

STEP 2

First, we will set up the equation that represents the total number of coins.x+y=37x + y = 37x+y=37This equation states that the sum of the number of nickels (xxx) and the number of quarters (yyy) is equal to the total number of coins, which is 37.

STEP 3

Next, we will set up the equation that represents the total value of the coins in dollars.0.05x+0.25y=6.250.05x + 0.25y = 6.250.05x+0.25y=6.25This equation states that the value of the nickels (5 cents each) plus the value of the quarters (25 cents each) equals $6.25.

STEP 4

Now we have a system of two equations with two variables:\begin{align*} x + y &= 37 \\ 0.05x + 0.25y &= 6.25 \end{align*}

STEP 5

We can express the system in matrix form as:[110.050.25][xy]=[376.25]\begin{bmatrix} 1 & 1 \\ 0.05 & 0.25 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 37 \\ 6.25 \end{bmatrix}[10.05​10.25​][xy​]=[376.25​]

STEP 6

To solve the system, we can use either substitution or elimination. For this problem, let's use substitution. We will solve the first equation for xxx:x=37−yx = 37 - yx=37−y

STEP 7

Now we will substitute x=37−yx = 37 - yx=37−y into the second equation:0.05(37−y)+0.25y=6.250.05(37 - y) + 0.25y = 6.250.05(37−y)+0.25y=6.25

STEP 8

Distribute 0.050.050.05 to both terms inside the parentheses:1.85−0.05y+0.25y=6.251.85 - 0.05y + 0.25y = 6.251.85−0.05y+0.25y=6.25

STEP 9

Combine like terms:1.85+0.20y=6.251.85 + 0.20y = 6.251.85+0.20y=6.25

STEP 10

Subtract 1.851.851.85 from both sides to isolate the term with yyy:0.20y=6.25−1.850.20y = 6.25 - 1.850.20y=6.25−1.85

STEP 11

Calculate the right side of the equation:0.20y=4.400.20y = 4.400.20y=4.40

STEP 12

Divide both sides by 0.200.200.20 to solve for yyy:y=4.400.20y = \frac{4.40}{0.20}y=0.204.40​

STEP 13

Calculate the value of yyy:y=22y = 22y=22

STEP 14

Now that we have the value of yyy, we can substitute it back into the equation x=37−yx = 37 - yx=37−y to find the value of xxx:x=37−22x = 37 - 22x=37−22

STEP 15

Calculate the value of xxx:x=15x = 15x=15So, we have x=15x = 15x=15 and y=22y = 22y=22. This means there are 15 nickels and 22 quarters.x:x:x: 15 nickels y:y:y: 22 quarters

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First, we will set up the equation that represents the total number of coins.
$x + y = 37$
This equation states that the sum of the number of nickels ( $x$ ) and the number of quarters ( $y$ ) is equal to the total number of coins, which is 37.

Next, we will set up the equation that represents the total value of the coins in dollars.
$0.05x + 0.25y = 6.25$
This equation states that the value of the nickels (5 cents each) plus the value of the quarters (25 cents each) equals $6.25.

Now we have a system of two equations with two variables:
\begin{align} x + y &= 37 \\ 0.05x + 0.25y &= 6.25 \end{align}

We can express the system in matrix form as:
$\begin{bmatrix} 1 & 1 \\ 0.05 & 0.25 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 37 \\ 6.25 \end{bmatrix}$

To solve the system, we can use either substitution or elimination. For this problem, let's use substitution. We will solve the first equation for $x$ :
$x = 37 - y$

Now we will substitute $x = 37 - y$ into the second equation:
$0.05(37 - y) + 0.25y = 6.25$

Distribute $0.05$ to both terms inside the parentheses:
$1.85 - 0.05y + 0.25y = 6.25$

Combine like terms:
$1.85 + 0.20y = 6.25$

Subtract $1.85$ from both sides to isolate the term with $y$ :
$0.20y = 6.25 - 1.85$

Calculate the right side of the equation:
$0.20y = 4.40$

Divide both sides by $0.20$ to solve for $y$ :
$y = \frac{4.40}{0.20}$

Calculate the value of $y$ :
$y = 22$

Now that we have the value of $y$ , we can substitute it back into the equation $x = 37 - y$ to find the value of $x$ :
$x = 37 - 22$

Calculate the value of $x$ :
$x = 15$
So, we have $x = 15$ and $y = 22$ . This means there are 15 nickels and 22 quarters.
$x:$ 15 nickels $y:$ 22 quarters