Abby is eight years older than Cari. Twenty years ago, Abby was three times as old as Cari. How old are they now?
Let's assume Cari's current age is x.
So, Abby's current age is x + 8 (as given, Abby is eight years older than Cari).
Twenty years ago, Cari's age was (x - 20) and Abby's age was (x + 8 - 20).
According to the problem, Abby's age 20 years ago was three times Cari's age 20 years ago.
So, we can write the equation:
x + 8 - 20 = 3*(x - 20)
x - 12 = 3x - 60
2x = 48
x = 24
Cari's current age is x = 24 years old.
Abby's current age is x + 8 = 24 + 8 = 32 years old.
Therefore, Cari is 24 years old and Abby is 32 years old. Answer: \boxed{24, 32}.
Three years ago, T was twice as old as J. In two years the sum of their ages will be 28 years. Find their present ages.
Let's assume T's current age is x and J's current age is y.
Three years ago, T's age was x - 3 and J's age was y - 3.
According to the problem, three years ago T was twice as old as J, so we can write the equation:
x - 3 = 2(y - 3)
Simplifying this equation, we get:
x - 3 = 2y - 6
x = 2y - 6 + 3
x = 2y - 3
In two years, T's age will be x + 2 and J's age will be y + 2.
According to the problem, the sum of their ages in two years will be 28, so we can write the equation:
(x + 2) + (y + 2) = 28
Simplifying this equation, we get:
x + y + 4 = 28
x + y = 28 - 4
x + y = 24
Now we have a system of equations:
x = 2y - 3
x + y = 24
We can solve this system of equations by substitution or elimination.
From the first equation, we can express x in terms of y:
x = 2y - 3
Substituting this expression for x in the second equation, we get:
(2y - 3) + y = 24
3y - 3 = 24
3y = 24 + 3
3y = 27
y = 27/3
y = 9
Then, substituting the value of y in the expression for x, we get:
x = 2(9) - 3
x = 18 - 3
x = 15
Therefore, T's present age is 15 years and J's present age is 9 years. Answer: \boxed{15, 9}.
The sum of the digits of a three digit number is 6. The hundreds is twice the units digit, and the tens digit is equal to the sum of the other two. Find the number.
Let's assume the number is written as XYZ, where X represents the hundreds digit, Y represents the tens digit, and Z represents the units digit.
According to the problem, the sum of the digits is 6:
X + Y + Z = 6
The hundreds digit is twice the units digit:
X = 2Z
The tens digit is equal to the sum of the other two:
Y = X + Z
Substituting the value of X from the second equation into the third equation, we get:
Y = 2Z + Z
Y = 3Z
Now we have a system of equations:
X + Y + Z = 6
X = 2Z
Y = 3Z
We can solve this system of equations by substitution.
From the second equation, we can express X in terms of Z:
X = 2Z
Substituting this expression for X in the first equation, we get:
2Z + Y + Z = 6
3Z + Y = 6
Substituting the expression for Y from the third equation in the last equation, we get:
3Z + 3Z = 6
6Z = 6
Z = 1
Now, substituting Z = 1 in the expression for Y, we get:
Y = 3(1) = 3
Finally, substituting Z = 1 in the expression for X, we get:
X = 2(1) = 2
Therefore, the number is 213. Answer: \boxed{213}.
The units digit is twice the tens digit. If the number is doubled, it will be twelve more than the reversed number. Find the number.
Let's assume the number is written as XYZ, where X represents the hundreds digit, Y represents the tens digit, and Z represents the units digit.
According to the problem, the units digit is twice the tens digit:
Z = 2Y
We are given that if the number is doubled, it will be twelve more than the reversed number. The reversed number will be ZYX:
2 * (100X + 10Y + Z) = 10Z + Y + X + 12
200X + 20Y + 2Z = 10Z + Y + X + 12
199X + 19Y - 8Z = 12
Now, let's substitute Z = 2Y into the equation:
199X + 19Y - 8(2Y) = 12
199X + 19Y - 16Y = 12
199X + 3Y = 12
Since we know the units digit is always less than or equal to 9, we can narrow down our possible values for Y. Let's test each value from 0 to 4:
If Y = 0, then 199X + 3(0) = 12
This results in X ≈ 0.0603, which is not an integer. Therefore, Y cannot be 0.
If Y = 1, then 199X + 3(1) = 12
This results in X ≈ -0.0653, which is not an integer. Therefore, Y cannot be 1.
If Y = 2, then 199X + 3(2) = 12
This results in X ≈ -0.1095, which is not an integer. Therefore, Y cannot be 2.
If Y = 3, then 199X + 3(3) = 12
This simplifies to 199X = 3, which does not result in an integer value for X. Therefore, Y cannot be 3.
If Y = 4, then 199X + 3(4) = 12
This simplifies to 199X = 0, which gives X = 0.
Therefore, the tens digit Y = 4 and the hundreds digit X = 0. From the equation Z = 2Y, we have Z = 8.
Therefore, the number is 048. Answer: \boxed{048}.
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To solve this problem, let's first assign variables to the ages of Abby and Cari.
Let's say Abby's current age is A, and Cari's current age is C.
According to the problem, Abby is eight years older than Cari. So we can write this relationship as:
A = C + 8 -- Equation 1
Twenty years ago, Abby was three times as old as Cari. So we can write this relationship as:
(A - 20) = 3*(C - 20) -- Equation 2
Now we have a system of two equations that we can solve to find the values of A and C.
To solve the system of equations, we can substitute Equation 1 into Equation 2.
(C + 8 - 20) = 3*(C - 20)
Now, let's simplify the equation:
C - 12 = 3C - 60
By combining like terms, the equation becomes:
-2C = -48
Dividing both sides of the equation by -2, we get:
C = 24
Now, we can substitute the value of C back into Equation 1 to find the value of A:
A = 24 + 8
A = 32
Therefore, Abby is currently 32 years old and Cari is currently 24 years old.