Help Math Help Bott

[Kennn03]

"A" can do a certain job in 4 hours, "B" can do the job in 6 hours, "C" can do the job in 8 hours. How long will it take to do the job if "A" and "B" work for 1 hour and "B" and "C" finish the job?

A tank can be filled by one pipe in 9 hours and another pipe in 12 hours. Starting empty, how long will it take to fill the tank if water is being taken out by a third pipe at a rate per hour equal to one - sixth the capacity of the tank?

 

[🤖]

For the first question:

Let's find the rate at which each person completes the job.

A can complete the job in 4 hours, so his rate is 1/4 of the job per hour.
B can complete the job in 6 hours, so his rate is 1/6 of the job per hour.
C can complete the job in 8 hours, so his rate is 1/8 of the job per hour.

If A and B work together for 1 hour, they complete (1/4 + 1/6) of the job, which is (3/12 + 2/12) = 5/12 of the job.

Now, we need to find how long it will take for B and C to finish the remaining job, which is 1 - 5/12 = 7/12 of the job.

Since B can complete 1/6 of the job per hour, and C can complete 1/8 of the job per hour, together they can complete (1/6 + 1/8) of the job per hour, which is (4/24 + 3/24) = 7/24 of the job per hour.

To find the time it takes for B and C to finish the remaining job (7/12), we can use the formula:

Time = Amount / Rate
Time = (7/12) / (7/24)
Time = (7/12) * (24/7)
Time = 2 hours

Therefore, it will take B and C 2 hours to finish the remaining job after A and B have worked for 1 hour.

For the second question:

Let's find the rate at which each pipe fills the tank.

The first pipe can fill the tank in 9 hours, so its rate is 1/9 of the tank per hour.
The second pipe can fill the tank in 12 hours, so its rate is 1/12 of the tank per hour.

The third pipe is taking out water from the tank at a rate of one-sixth of the tank's capacity per hour, which is 1/6 of the tank per hour.

To find the combined rate at which water is being added or taken out, we need to subtract the rate of the third pipe from the combined rate of the first two *****.

Combined rate = (1/9 + 1/12) - 1/6
Combined rate = (4/36 + 3/36) - 6/36
Combined rate = (7/36) - (6/36)
Combined rate = 1/36 of the tank per hour

Now, to find the time it takes to fill the tank, we can use the formula:

Time = Capacity / Rate
Time = 1 / (1/36)
Time = 36 hours

Therefore, it will take 36 hours to fill the tank if water is being taken out by the third pipe at a rate per hour equal to one-sixth the capacity of the tank.

 

[Kennn03]

For the first question:

Let's find the rate at which each person completes the job.

A can complete the job in 4 hours, so his rate is 1/4 of the job per hour.
B can complete the job in 6 hours, so his rate is 1/6 of the job per hour.
C can complete the job in 8 hours, so his rate is 1/8 of the job per hour.

If A and B work together for 1 hour, they complete (1/4 + 1/6) of the job, which is (3/12 + 2/12) = 5/12 of the job.

Now, we need to find how long it will take for B and C to finish the remaining job, which is 1 - 5/12 = 7/12 of the job.

Since B can complete 1/6 of the job per hour, and C can complete 1/8 of the job per hour, together they can complete (1/6 + 1/8) of the job per hour, which is (4/24 + 3/24) = 7/24 of the job per hour.

To find the time it takes for B and C to finish the remaining job (7/12), we can use the formula:

Time = Amount / Rate
Time = (7/12) / (7/24)
Time = (7/12) * (24/7)
Time = 2 hours

Therefore, it will take B and C 2 hours to finish the remaining job after A and B have worked for 1 hour.

For the second question:

Let's find the rate at which each pipe fills the tank.

The first pipe can fill the tank in 9 hours, so its rate is 1/9 of the tank per hour.
The second pipe can fill the tank in 12 hours, so its rate is 1/12 of the tank per hour.

The third pipe is taking out water from the tank at a rate of one-sixth of the tank's capacity per hour, which is 1/6 of the tank per hour.

To find the combined rate at which water is being added or taken out, we need to subtract the rate of the third pipe from the combined rate of the first two *****.

Combined rate = (1/9 + 1/12) - 1/6
Combined rate = (4/36 + 3/36) - 6/36
Combined rate = (7/36) - (6/36)
Combined rate = 1/36 of the tank per hour

Now, to find the time it takes to fill the tank, we can use the formula:

Time = Capacity / Rate
Time = 1 / (1/36)
Time = 36 hours

Therefore, it will take 36 hours to fill the tank if water is being taken out by the third pipe at a rate per hour equal to one-sixth the capacity of the tank.

bot also answer these questions:

How many ounces of pure silver must be added to 100 ounces, 40% pure, to make an alloy which is 65% pure silver?

A perfumer wishes to blend perfume valued at Php 4.10 an ounce with perfume worth Php 2.50 an ounce to obtain a mixture of 40 ounces worth Php 3.00 an ounce. How much of the Php 4.10 perfume should use?

A gourmet chef blends a salad dressing by mixing 20 ounces of a solution containing 85% olive oil with pure corn oil, in order that the dressing be 50% olive oil. How much corn oil should be use?

What percentage of a mixture of sand, gravel and cement containing 30% cement should be replaced by pure cement in order to produce that is 40% cement?

 

[🤖]

bot also answer these questions:

How many ounces of pure silver must be added to 100 ounces, 40% pure, to make an alloy which is 65% pure silver?

A perfumer wishes to blend perfume valued at Php 4.10 an ounce with perfume worth Php 2.50 an ounce to obtain a mixture of 40 ounces worth Php 3.00 an ounce. How much of the Php 4.10 perfume should use?

A gourmet chef blends a salad dressing by mixing 20 ounces of a solution containing 85% olive oil with pure corn oil, in order that the dressing be 50% olive oil. How much corn oil should be use?

What percentage of a mixture of sand, gravel and cement containing 30% cement should be replaced by pure cement in order to produce that is 40% cement?

For the first question:
To find how many ounces of pure silver must be added to 100 ounces, 40% pure, to make an alloy which is 65% pure silver, we can set up an equation based on the amount of pure silver in the initial mixture and the final mixture.

Let x be the number of ounces of pure silver to be added.

In the initial mixture of 100 ounces, 40% of it is pure silver, so we have 0.40 * 100 = 40 ounces of pure silver.

In the final mixture, we have 100 + x total ounces, and we want it to be 65% pure silver. So, we have 0.65 * (100 + x) ounces of pure silver.

Equating the amount of pure silver in the initial and final mixtures, we can write the equation:

40 + x = 0.65 * (100 + x)

Expanding the equation, we get:

40 + x = 65 + 0.65x

Combining like terms, we get:

0.35x = 25

Dividing both sides by 0.35, we find:

x = 71.43

Therefore, approximately 71.43 ounces of pure silver must be added to the initial 100 ounces, 40% pure, to make an alloy which is 65% pure silver.

For the second question:
To determine how much of the Php 4.10 perfume should be used to blend with the Php 2.50 perfume in order to obtain a mixture of 40 ounces worth Php 3.00 an ounce, we can set up an equation based on the value and quantity of the perfumes.

Let x be the amount of the Php 4.10 perfume to be used.

The value of the Php 4.10 perfume used is 4.10x.
The value of the Php 2.50 perfume used is 2.50(40 - x), since the total quantity is 40 ounces and we are using x ounces of the more expensive perfume.

The total value of the mixture is 3.00 * 40 = 120.

Setting up the equation, we have:

4.10x + 2.50(40 - x) = 120

Expanding and simplifying, we get:

4.10x + 100 - 2.50x = 120

Combining like terms, we have:

1.60x = 20

Dividing both sides by 1.60, we find:

x ≈ 12.5

Therefore, approximately 12.5 ounces of the Php 4.10 perfume should be used in order to obtain the desired mixture.

For the third question:
To find out how much corn oil should be used when mixing 20 ounces of a solution containing 85% olive oil, in order to make a dressing that is 50% olive oil, we can set up an equation based on the amount of olive oil in the initial solution and the desired mixture.

Let x be the amount of corn oil to be used.

In the initial solution of 20 ounces, 85% of it is olive oil, so we have 0.85 * 20 = 17 ounces of olive oil.

In the final mixture, we have 20 + x total ounces, and we want it to be 50% olive oil. So, we have 0.50 * (20 + x) ounces of olive oil.

Equating the amount of olive oil in the initial and final mixtures, we can write the equation:

17 = 0.50 * (20 + x)

Expanding the equation, we get:

17 = 10 + 0.50x

Subtracting 10 from both sides, we have:

7 = 0.50x

Dividing both sides by 0.50, we find:

x = 14

Therefore, 14 ounces of corn oil should be used to make the desired dressing.

For the fourth question:
To determine the percentage of the mixture of sand, gravel, and cement that should be replaced by pure cement in order to produce a mixture that is 40% cement, we need to set up an equation based on the amount of cement in the initial mixture and the desired mixture.

Let x be the percentage of the mixture to be replaced by pure cement.

In the initial mixture, 30% of it is cement, so we have 0.30x of cement.

In the final mixture, we want it to be 40% cement, so we have 0.40(100 - x) of cement.

Equating the amount of cement in the initial and final mixtures, we can write the equation:

0.30x = 0.40(100 - x)

Expanding the equation, we get:

0.30x = 40 - 0.40x

Combining like terms, we have:

0.70x = 40

Dividing both sides by 0.70, we find:

x ≈ 57.14

Therefore, approximately 57.14% of the mixture of sand, gravel, and cement should be replaced by pure cement in order to produce a mixture that is 40% cement.