Question Find the number of tongue depressors and Band-Aids in a medic's bag worth $13.14 with 183 total packages, where tongue depressors cost$ 0.05 and Band-Aids cost $0.08.$ \# \text{ of Tongue Depressors}\# \text{ of Band-Aids}$

Studdy Solution

STEP 1

Assumptions
1. The cost of one tongue depressor is $0.05.<br />2. The cost of one Band-Aid is$ 0.08.
3. The total value of the materials is $13.14.
4. There are 183 individual packages in total, which include both tongue depressors and Band-Aids.
5. We are looking for the number of tongue depressors and Band-Aids.

STEP 2

Let's denote the number of tongue depressors as $x$ and the number of Band-Aids as $y$ . We can then set up two equations based on the given information.

STEP 3

The first equation will represent the total number of packages:
$x + y = 183$

STEP 4

The second equation will represent the total value of the materials:
$0.05x + 0.08y = 13.14$

STEP 5

We now have a system of linear equations:
\begin{align} x + y &= 183 \\ 0.05x + 0.08y &= 13.14 \end{align}

STEP 6

To solve this system, we can use the method of substitution or elimination. Let's use the elimination method.

STEP 7

First, we'll multiply the first equation by $0.05$ to align the coefficient of $x$ with the second equation:
$0.05(x + y) = 0.05(183)$

STEP 8

Simplify the multiplication:
$0.05x + 0.05y = 9.15$

STEP 9

Now we have a new system of equations:
\begin{align} 0.05x + 0.05y &= 9.15 \\ 0.05x + 0.08y &= 13.14 \end{align}

STEP 10

Subtract the first equation from the second equation to eliminate $x$ :
$(0.05x + 0.08y) - (0.05x + 0.05y) = 13.14 - 9.15$

STEP 11

Simplify the subtraction:
$0.03y = 3.99$

STEP 12

Divide both sides by $0.03$ to solve for $y$ :
$y = \frac{3.99}{0.03}$

STEP 13

Calculate the value of $y$ :
$y = \frac{3.99}{0.03} = 133$

STEP 14

Now that we have the value of $y$ , which represents the number of Band-Aids, we can substitute this value into the first equation to solve for $x$ :
$x + 133 = 183$

STEP 15

Subtract 133 from both sides to solve for $x$ :
$x = 183 - 133$

STEP 16

Calculate the value of $x$ :
$x = 183 - 133 = 50$

STEP 17

We have found the number of tongue depressors and Band-Aids:
\# of Tongue Depressors: $x = 50$ \# of Band-Aids: $y = 133$

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Question Find the number of tongue depressors and Band-Aids in a medic's bag worth $13.14 with 183 total packages, where tongue depressors cost$ 0.05 and Band-Aids cost $0.08.$ \# \text{ of Tongue Depressors}\# \text{ of Band-Aids}$

Studdy Solution

STEP 1

STEP 2

Let's denote the number of tongue depressors as $x$ and the number of Band-Aids as $y$ . We can then set up two equations based on the given information.

STEP 3

The first equation will represent the total number of packages:
$x + y = 183$

STEP 4

The second equation will represent the total value of the materials:
$0.05x + 0.08y = 13.14$

STEP 5

We now have a system of linear equations:
\begin{align} x + y &= 183 \\ 0.05x + 0.08y &= 13.14 \end{align}

STEP 6

To solve this system, we can use the method of substitution or elimination. Let's use the elimination method.

STEP 7

First, we'll multiply the first equation by $0.05$ to align the coefficient of $x$ with the second equation:
$0.05(x + y) = 0.05(183)$

STEP 8

Simplify the multiplication:
$0.05x + 0.05y = 9.15$

STEP 9

Now we have a new system of equations:
\begin{align} 0.05x + 0.05y &= 9.15 \\ 0.05x + 0.08y &= 13.14 \end{align}

STEP 10

Subtract the first equation from the second equation to eliminate $x$ :
$(0.05x + 0.08y) - (0.05x + 0.05y) = 13.14 - 9.15$

STEP 11

Simplify the subtraction:
$0.03y = 3.99$

STEP 12

Divide both sides by $0.03$ to solve for $y$ :
$y = \frac{3.99}{0.03}$

STEP 13

Calculate the value of $y$ :
$y = \frac{3.99}{0.03} = 133$

STEP 14

Now that we have the value of $y$ , which represents the number of Band-Aids, we can substitute this value into the first equation to solve for $x$ :
$x + 133 = 183$

STEP 15

Subtract 133 from both sides to solve for $x$ :
$x = 183 - 133$

STEP 16

Calculate the value of $x$ :
$x = 183 - 133 = 50$

STEP 17

We have found the number of tongue depressors and Band-Aids:
\# of Tongue Depressors: $x = 50$ \# of Band-Aids: $y = 133$

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Studdy Solution

STEP 1

STEP 2

Let's denote the number of tongue depressors as xxx and the number of Band-Aids as yyy. We can then set up two equations based on the given information.

STEP 3

The first equation will represent the total number of packages:x+y=183x + y = 183x+y=183

STEP 4

The second equation will represent the total value of the materials:0.05x+0.08y=13.140.05x + 0.08y = 13.140.05x+0.08y=13.14

STEP 5

We now have a system of linear equations:\begin{align*} x + y &= 183 \\ 0.05x + 0.08y &= 13.14 \end{align*}

STEP 6

To solve this system, we can use the method of substitution or elimination. Let's use the elimination method.

STEP 7

First, we'll multiply the first equation by 0.050.050.05 to align the coefficient of xxx with the second equation:0.05(x+y)=0.05(183)0.05(x + y) = 0.05(183)0.05(x+y)=0.05(183)

STEP 8

Simplify the multiplication:0.05x+0.05y=9.150.05x + 0.05y = 9.150.05x+0.05y=9.15

STEP 9

Now we have a new system of equations:\begin{align*} 0.05x + 0.05y &= 9.15 \\ 0.05x + 0.08y &= 13.14 \end{align*}

STEP 10

Subtract the first equation from the second equation to eliminate xxx:(0.05x+0.08y)−(0.05x+0.05y)=13.14−9.15(0.05x + 0.08y) - (0.05x + 0.05y) = 13.14 - 9.15(0.05x+0.08y)−(0.05x+0.05y)=13.14−9.15

STEP 11

Simplify the subtraction:0.03y=3.990.03y = 3.990.03y=3.99

STEP 12

Divide both sides by 0.030.030.03 to solve for yyy:y=3.990.03y = \frac{3.99}{0.03}y=0.033.99​

STEP 13

Calculate the value of yyy:y=3.990.03=133y = \frac{3.99}{0.03} = 133y=0.033.99​=133

STEP 14

Now that we have the value of yyy, which represents the number of Band-Aids, we can substitute this value into the first equation to solve for xxx:x+133=183x + 133 = 183x+133=183

STEP 15

Subtract 133 from both sides to solve for xxx:x=183−133x = 183 - 133x=183−133

STEP 16

Calculate the value of xxx:x=183−133=50x = 183 - 133 = 50x=183−133=50

STEP 17

We have found the number of tongue depressors and Band-Aids:\# of Tongue Depressors: x=50x = 50x=50 \# of Band-Aids: y=133y = 133y=133

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Let's denote the number of tongue depressors as $x$ and the number of Band-Aids as $y$ . We can then set up two equations based on the given information.

The first equation will represent the total number of packages:
$x + y = 183$

The second equation will represent the total value of the materials:
$0.05x + 0.08y = 13.14$

We now have a system of linear equations:
\begin{align} x + y &= 183 \\ 0.05x + 0.08y &= 13.14 \end{align}

First, we'll multiply the first equation by $0.05$ to align the coefficient of $x$ with the second equation:
$0.05(x + y) = 0.05(183)$

Simplify the multiplication:
$0.05x + 0.05y = 9.15$

Now we have a new system of equations:
\begin{align} 0.05x + 0.05y &= 9.15 \\ 0.05x + 0.08y &= 13.14 \end{align}

Subtract the first equation from the second equation to eliminate $x$ :
$(0.05x + 0.08y) - (0.05x + 0.05y) = 13.14 - 9.15$

Simplify the subtraction:
$0.03y = 3.99$

Divide both sides by $0.03$ to solve for $y$ :
$y = \frac{3.99}{0.03}$

Calculate the value of $y$ :
$y = \frac{3.99}{0.03} = 133$

Now that we have the value of $y$ , which represents the number of Band-Aids, we can substitute this value into the first equation to solve for $x$ :
$x + 133 = 183$

Subtract 133 from both sides to solve for $x$ :
$x = 183 - 133$

Calculate the value of $x$ :
$x = 183 - 133 = 50$

We have found the number of tongue depressors and Band-Aids:
\# of Tongue Depressors: $x = 50$ \# of Band-Aids: $y = 133$