Hi!
My name is Jason,
I needed help studying for tomorrow's test.
The exam is on number problems.
EX. One positive number is 4 times another. IF 240 is divided by each number, the greater quotient exceeds the lesser by 15. Find the two numbers.
Also, there will be some problems on work.
EX. It takes 6 minutes to fill a certain pool and 18 min. to drain the same poll when it is full. With the drain open and the pool empty, how long would it take to fill the pool?
I really need a fast, easy way to solve these types of problems. If you have suggestions, please help out.
Pick some variables to represent what you are trying to find.
Here, let:
X = smaller positive number
Y = larger positive number
Algebraically express all the relationships given by the problem.
Y = 4X
(240 / X) - 15 = (240 / Y)
It usually seems to simplify things if you get rid of variables in the denominators of fractions. If both sides of the second equation are multiplied by X * Y it yields:
240Y - 15XY = 240X.
Substitute the value for Y from the first equation.
240(4X) - 15X(4X) = 240X
960X - 60X^2 = 240X
divide both sides by 60X
16 - X = 4
16 - 4 = X
X = 12
Y = 4X
Y = 48
Hi Jason! I'd be happy to help you study for your test on number problems. Let's go through each problem one by one and I'll explain the steps to solve them.
Problem 1: One positive number is 4 times another. If 240 is divided by each number, the greater quotient exceeds the lesser by 15. Find the two numbers.
Step 1: Let's assign variables to the unknown numbers. Let's say the smaller number is x and the larger number is 4x (since one number is 4 times the other).
Step 2: Divide 240 by each number:
240 ÷ x = quotient for the smaller number
240 ÷ (4x) = quotient for the larger number
Step 3: The problem states that the greater quotient exceeds the lesser by 15. So we can set up an equation:
240 ÷ (4x) - 240 ÷ x = 15
Step 4: Solve the equation by simplifying and solving for x:
60 - 15 = 240 ÷ x
45 = 240 ÷ x
x = 240 ÷ 45
To find the larger number, substitute the value of x back into the expression 4x:
4x = 4 * (240 ÷ 45)
The answer would be the two numbers you found.
Now let's move on to the second problem.
Problem 2: It takes 6 minutes to fill a certain pool and 18 minutes to drain the same pool when it is full. With the drain open and the pool empty, how long would it take to fill the pool?
Step 1: Let's assign variables to the rates at which the pool fills and drains. Let's say the rate of filling is f (in units per minute) and the rate of draining is d (in units per minute).
Step 2: The filling rate and draining rate can be found by taking the reciprocal of the time it takes to fill or drain the pool:
Filling rate: 1 / 6 (since it takes 6 minutes to fill the pool)
Draining rate: 1 / 18 (since it takes 18 minutes to drain the pool)
Step 3: Subtract the draining rate from the filling rate to find the combined rate:
Combined rate = filling rate - draining rate
Combined rate = 1 / 6 - 1 / 18
Step 4: Find the reciprocal of the combined rate to determine the time it takes to fill the pool when the drain is open and the pool is empty:
Time to fill pool = 1 / (1 / 6 - 1 / 18)
Simplify the expression to get the answer.
I hope this helps! Remember to practice solving similar problems to improve your skills. Let me know if you have any further questions.