Four years ago, Danielle was 2/3 as old as Jim was. Now she is 3/4 as old as he is. How old is each now?
Jim's present age ---- x
Danielle's present age ----- 3x/4
4 years ago:
Jim ---- x-4
Danielle ----- 3x/4 - 4
3x/4 - 4 = (2/3)(x-4)
3x/4 - 4 = 2x/3 - 8/3
multiply each term by 12
9x - 48 = 8x - 32
x = 16
Jim is now 16, and Danielle now is 12
check:
4 years ago Jim was 12 and Danielle was 8
is 2/3 of 12 equal to 8? yes!
D - 4 = 2/3 (J - 4) ...3 D - 12 = 2 J - 8
D = 3/4 J
substituting ... 9/4 J = 8/4 J + 4
solve for J , then substitute back to find D
To solve this problem, let's assign variables to represent the ages of Danielle and Jim. Let's say Danielle's current age is D, and Jim's current age is J.
From the information given, we can create two equations:
Equation 1: Four years ago, Danielle was 2/3 as old as Jim was.
This can be written as: D - 4 = (2/3)(J - 4)
Equation 2: Now she is 3/4 as old as he is.
This can be written as: D = (3/4)J
Now we can solve this system of equations to find the values of D and J.
Step 1: Simplify Equation 1
D - 4 = (2/3)(J - 4)
Multiply both sides of the equation by 3 to eliminate the fraction:
3(D - 4) = 2(J - 4)
Expand the equation:
3D - 12 = 2J - 8
Step 2: Simplify Equation 2
D = (3/4)J
Step 3: Solve for J in terms of D in Equation 2
From Equation 2, we can isolate J:
Multiply both sides of the equation by 4 to eliminate the fraction:
4D = 3J
Divide both sides of the equation by 3:
J = (4/3)D
Step 4: Substitute J in Equation 3
Substitute (4/3)D for J in Equation 1:
3D - 12 = 2((4/3)D) - 8
Step 5: Solve for D
Distribute 2 to (4/3)D:
3D - 12 = (8/3)D - 8
Multiply both sides of the equation by 3 to eliminate the fraction:
9D - 36 = 8D - 24
Combine like terms:
9D - 8D = -24 + 36
D = 12
Step 6: Calculate J using D
Substitute D = 12 into Equation 1 or 2:
J = (4/3)D = (4/3)(12) = 16
Therefore, Danielle is currently 12 years old, and Jim is currently 16 years old.