Solved on Feb 18, 2024

Find the time taken by the newer computer to send the email alone, given that the older computer takes twice as long and the two together take 15 minutes.
Solution: Let $x$ be the time taken by the newer computer. Then the time taken by the older computer is $2x$ . The two computers together take 15 minutes, so $\frac{1}{x} + \frac{1}{2x} = \frac{1}{15}$ . Solving for $x$ , we get $x = 10$ minutes.

STEP 1

Assumptions
1. The older computer takes twice as long as the newer computer to send out the email.
2. When working together, the two computers take 15 minutes to send out the email.
3. We need to find the time it takes for the newer computer to send out the email on its own.

STEP 2

Let's denote the time it takes for the newer computer to send out the email on its own as $x$ minutes.

STEP 3

According to the given information, the older computer takes twice as long to send out the email. Therefore, it takes the older computer $2x$ minutes to send out the email on its own.

STEP 4

The rate at which the newer computer works is $\frac{1}{x}$ of the job per minute, since it completes 1 job in $x$ minutes.

STEP 5

Similarly, the rate at which the older computer works is $\frac{1}{2x}$ of the job per minute, since it completes 1 job in $2x$ minutes.

STEP 6

When working together, their combined rate is the sum of their individual rates. Therefore, the combined rate is $\frac{1}{x} + \frac{1}{2x}$ .

STEP 7

We know that working together, they take 15 minutes to send out the email. This means their combined rate accomplishes 1 job in 15 minutes, so the combined rate is $\frac{1}{15}$ of the job per minute.

STEP 8

Now we can set up the equation based on their combined rate:
$\frac{1}{x} + \frac{1}{2x} = \frac{1}{15}$

STEP 9

To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator is $2x$ .

STEP 10

Rewrite the equation with the common denominator:
$\frac{2}{2x} + \frac{1}{2x} = \frac{1}{15}$

STEP 11

Combine the fractions on the left side of the equation:
$\frac{2 + 1}{2x} = \frac{1}{15}$

STEP 12

Simplify the numerator on the left side of the equation:
$\frac{3}{2x} = \frac{1}{15}$

STEP 13

To find the value of $x$ , we can cross-multiply:
$3 \cdot 15 = 2x \cdot 1$

STEP 14

Perform the multiplication:
$45 = 2x$

STEP 15

Divide both sides of the equation by 2 to solve for $x$ :
$x = \frac{45}{2}$

STEP 16

Calculate the value of $x$ :
$x = 22.5$
The newer computer takes 22.5 minutes to send out the email on its own.

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STEP 1

Assumptions
1. The older computer takes twice as long as the newer computer to send out the email.
2. When working together, the two computers take 15 minutes to send out the email.
3. We need to find the time it takes for the newer computer to send out the email on its own.

STEP 2

Let's denote the time it takes for the newer computer to send out the email on its own as $x$ minutes.

STEP 3

According to the given information, the older computer takes twice as long to send out the email. Therefore, it takes the older computer $2x$ minutes to send out the email on its own.

STEP 4

The rate at which the newer computer works is $\frac{1}{x}$ of the job per minute, since it completes 1 job in $x$ minutes.

STEP 5

Similarly, the rate at which the older computer works is $\frac{1}{2x}$ of the job per minute, since it completes 1 job in $2x$ minutes.

STEP 6

When working together, their combined rate is the sum of their individual rates. Therefore, the combined rate is $\frac{1}{x} + \frac{1}{2x}$ .

STEP 7

We know that working together, they take 15 minutes to send out the email. This means their combined rate accomplishes 1 job in 15 minutes, so the combined rate is $\frac{1}{15}$ of the job per minute.

STEP 8

Now we can set up the equation based on their combined rate:
$\frac{1}{x} + \frac{1}{2x} = \frac{1}{15}$

STEP 9

To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator is $2x$ .

STEP 10

Rewrite the equation with the common denominator:
$\frac{2}{2x} + \frac{1}{2x} = \frac{1}{15}$

STEP 11

Combine the fractions on the left side of the equation:
$\frac{2 + 1}{2x} = \frac{1}{15}$

STEP 12

Simplify the numerator on the left side of the equation:
$\frac{3}{2x} = \frac{1}{15}$

STEP 13

To find the value of $x$ , we can cross-multiply:
$3 \cdot 15 = 2x \cdot 1$

STEP 14

Perform the multiplication:
$45 = 2x$

STEP 15

Divide both sides of the equation by 2 to solve for $x$ :
$x = \frac{45}{2}$

STEP 16

Calculate the value of $x$ :
$x = 22.5$
The newer computer takes 22.5 minutes to send out the email on its own.

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STEP 1

Assumptions1. The older computer takes twice as long as the newer computer to send out the email.2. When working together, the two computers take 15 minutes to send out the email.3. We need to find the time it takes for the newer computer to send out the email on its own.

STEP 2

Let's denote the time it takes for the newer computer to send out the email on its own as x x x minutes.

STEP 3

According to the given information, the older computer takes twice as long to send out the email. Therefore, it takes the older computer 2x 2x 2x minutes to send out the email on its own.

STEP 4

The rate at which the newer computer works is 1x \frac{1}{x} x1​ of the job per minute, since it completes 1 job in x x x minutes.

STEP 5

Similarly, the rate at which the older computer works is 12x \frac{1}{2x} 2x1​ of the job per minute, since it completes 1 job in 2x 2x 2x minutes.

STEP 6

When working together, their combined rate is the sum of their individual rates. Therefore, the combined rate is 1x+12x \frac{1}{x} + \frac{1}{2x} x1​+2x1​.

STEP 7

We know that working together, they take 15 minutes to send out the email. This means their combined rate accomplishes 1 job in 15 minutes, so the combined rate is 115 \frac{1}{15} 151​ of the job per minute.

STEP 8

Now we can set up the equation based on their combined rate:1x+12x=115 \frac{1}{x} + \frac{1}{2x} = \frac{1}{15} x1​+2x1​=151​

STEP 9

To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator is 2x 2x 2x.

STEP 10

Rewrite the equation with the common denominator:22x+12x=115 \frac{2}{2x} + \frac{1}{2x} = \frac{1}{15} 2x2​+2x1​=151​

STEP 11

Combine the fractions on the left side of the equation:2+12x=115 \frac{2 + 1}{2x} = \frac{1}{15} 2x2+1​=151​

STEP 12

Simplify the numerator on the left side of the equation:32x=115 \frac{3}{2x} = \frac{1}{15} 2x3​=151​

STEP 13

To find the value of x x x, we can cross-multiply:3⋅15=2x⋅1 3 \cdot 15 = 2x \cdot 1 3⋅15=2x⋅1

STEP 14

Perform the multiplication:45=2x 45 = 2x 45=2x

STEP 15

Divide both sides of the equation by 2 to solve for x x x:x=452 x = \frac{45}{2} x=245​

STEP 16

Calculate the value of x x x:x=22.5 x = 22.5 x=22.5The newer computer takes 22.5 minutes to send out the email on its own.

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Assumptions
1. The older computer takes twice as long as the newer computer to send out the email.
2. When working together, the two computers take 15 minutes to send out the email.
3. We need to find the time it takes for the newer computer to send out the email on its own.

Let's denote the time it takes for the newer computer to send out the email on its own as $x$ minutes.

According to the given information, the older computer takes twice as long to send out the email. Therefore, it takes the older computer $2x$ minutes to send out the email on its own.

The rate at which the newer computer works is $\frac{1}{x}$ of the job per minute, since it completes 1 job in $x$ minutes.

Similarly, the rate at which the older computer works is $\frac{1}{2x}$ of the job per minute, since it completes 1 job in $2x$ minutes.

When working together, their combined rate is the sum of their individual rates. Therefore, the combined rate is $\frac{1}{x} + \frac{1}{2x}$ .

We know that working together, they take 15 minutes to send out the email. This means their combined rate accomplishes 1 job in 15 minutes, so the combined rate is $\frac{1}{15}$ of the job per minute.

Now we can set up the equation based on their combined rate:
$\frac{1}{x} + \frac{1}{2x} = \frac{1}{15}$

To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator is $2x$ .

Rewrite the equation with the common denominator:
$\frac{2}{2x} + \frac{1}{2x} = \frac{1}{15}$

Combine the fractions on the left side of the equation:
$\frac{2 + 1}{2x} = \frac{1}{15}$

Simplify the numerator on the left side of the equation:
$\frac{3}{2x} = \frac{1}{15}$

To find the value of $x$ , we can cross-multiply:
$3 \cdot 15 = 2x \cdot 1$

Perform the multiplication:
$45 = 2x$

Divide both sides of the equation by 2 to solve for $x$ :
$x = \frac{45}{2}$

Calculate the value of $x$ :
$x = 22.5$
The newer computer takes 22.5 minutes to send out the email on its own.