As of today, Jeff's age is three times Ming's age. Tina's age is four more than two times Ming's age. Next year, Jeff's age will be seven more than two times Ming's age.
A. As of today, how many times as old as Ming is Tina? Justify your response
B. Let R(x) represent the ratio of Tina's age in years to Ming's age in years, and let x represent the number of years from now, either past or future. Write R as a function of x.
I really don't even know where to start!
Thank you for your help!
Just translate the English into Math
Ming's age --- x
Jeffs age is 3x
Tina's age = 2x+4
next year:
Jeff's age = 3x+1
Ming's age = x+1
It said: 3x+1 = 2(x+1) + 7
3x+1 = 2x+2+7
x = 10
So today:
Ming is 10
Jeff is 30
Tina is 24
A. So it looks like today, Tina is 2.4 times as old as Ming
B. R(x) = (24+x)/(10+x)
To solve this problem, let's start by assigning variables to the ages of Jeff, Ming, and Tina.
Let's say Jeff's age is J, Ming's age is M, and Tina's age is T.
According to the given information:
1. "As of today, Jeff's age is three times Ming's age":
J = 3M.
2. "Tina's age is four more than two times Ming's age":
T = 2M + 4.
3. "Next year, Jeff's age will be seven more than two times Ming's age":
J + 1 = 2(M + 1) + 7.
Now, let's solve each part of the problem.
A. As of today, how many times as old as Ming is Tina? Justify your response.
To find this, we need to compare Tina's age to Ming's age.
Substitute the value of T from equation 2 into equation 1:
2M + 4 = 3M.
Rearrange the equation to isolate Ming's age:
3M - 2M = 4,
M = 4.
Substitute the value of Ming's age back into equation 2 to find Tina's age:
T = 2(4) + 4 = 12.
Therefore, Tina is 12 years old and Ming is 4 years old.
To find how many times as old as Ming Tina is, we divide their ages:
T/M = 12/4 = 3.
So, Tina is three times as old as Ming as of today.
B. Let R(x) represent the ratio of Tina's age in years to Ming's age in years, and let x represent the number of years from now, either past or future. Write R as a function of x.
To find the ratio of their ages as a function of the number of years from now, we need to consider the information from equation 3.
Substitute the value of J from equation 1 into equation 3:
3M + 1 = 2(M + 1) + 7.
Simplify the equation:
3M + 1 = 2M + 2 + 7,
3M + 1 = 2M + 9.
Rearrange the equation to isolate Ming's age:
3M - 2M = 9 - 1,
M = 8.
Now, the ratio R(x) can be represented as the ratio of (T + x) to (M + x):
R(x) = (2M + 4 + x) / (M + x).
Therefore, R as a function of x is:
R(x) = (2(8) + 4 + x) / (8 + x).
Simplifying further:
R(x) = (16 + 4 + x) / (8 + x).
R(x) = (20 + x) / (8 + x).
So, the ratio of Tina's age to Ming's age after x years is (20 + x) / (8 + x).
To solve this problem, let's break it down step by step.
First, let's assign variables to the ages of Jeff (J), Ming (M), and Tina (T).
Based on the information given:
1. As of today, Jeff's age is three times Ming's age, so we can write J = 3M.
2. Tina's age is four more than two times Ming's age, so we can write T = 2M + 4.
3. Next year, Jeff's age will be seven more than two times Ming's age, so we can write J + 1 = 2M + 7.
Now, let's solve the problem.
A. To find out how many times older Tina is than Ming, we need to calculate the ratio of their ages. So we can write:
T/M = (2M + 4)/M
Simplifying this expression, we get:
T/M = 2 + 4/M
Therefore, as of today, Tina is "2 + 4/M" times older than Ming. To get the exact value, we would need to know Ming's age.
B. To write the ratio of Tina's age to Ming's age as a function of x (number of years from now), we need to consider the given information about Jeff's age. Let's rewrite Jeff's age equation as:
J = 3M
Next year, Jeff's age will be:
J + 1 = 3M + 1
According to the given information, Jeff's age next year will be seven more than two times Ming's age. So we can write:
J + 1 = 2M + 7
Now, let's rearrange this equation to solve it for M:
3M + 1 = 2M + 7
Subtracting 2M from both sides:
M + 1 = 7
Subtracting 1 from both sides:
M = 6
Therefore, as of today, Ming's age is 6 years. Now, we can rewrite Tina's current age equation:
T = 2M + 4
Substituting the value of M:
T = 2(6) + 4
T = 12 + 4
T = 16
So, as of today, Tina is 16 years old.
Now, to write the ratio of Tina's age (T) to Ming's age (M) as a function of x, we will use the equation:
R(x) = (T + x) / (M + x)
For example, if we want to know the ratio of their ages 5 years from now, we can substitute x = 5 into the equation:
R(5) = (T + 5) / (M + 5)
Simplifying this expression would give us the ratio of their ages 5 years from now.