Answer:
Bart is currently 32, and Art is currently 40.
Step-by-step explanation:
Step 1: Derive an Equation
Since Bart and Art's current ages are in a ratio of [tex]4:5[/tex], we can let their current ages be [tex]4x[/tex] and [tex]5x[/tex], respectively.
In 4 years, their ages will increase by 4. This means that in terms of [tex]x[/tex], Bart and Art will be [tex]4x+4[/tex] and [tex]5x+4[/tex] years old in 4 years, respectively.
We are given that their ages in 4 years will be in the ratio [tex]9:11[/tex]. Therefore, the ratios [tex]4x+4:5x+4[/tex] and [tex]9:11[/tex] are equivalent, so [tex]4x+4:5x+4 = 9:11[/tex].
Ratios can also be written as fractions, so we can derive the following equation:
[tex]\frac{4x+4}{5x+4}=\frac{9}{11}[/tex]
Step 2: Solve the Equation
Solving for [tex]x[/tex], we get:
[tex]\frac{4x+4}{5x+4}=\frac{9}{11}[/tex]
[tex](4x+4)*11=(5x+4)*9[/tex] (Cross-Multiply)
[tex]44x+44=45x+36[/tex] (Distribute)
[tex]44x+44-44x-36=45x+36-44x-36[/tex] (Subtract [tex]44x[/tex] and [tex]36[/tex] from both sides of the equation to isolate [tex]x[/tex])
[tex]8=x \rightarrow x=8[/tex] (Simplify / Symmetric Property of Equality)
Step 3: Answer the Question
We know that [tex]x=8[/tex], therefore, Bart is currently [tex]4*8=\bf 32[/tex] years old and Art is currently [tex]5*8=\bf 40[/tex] years old. Hope this helps!
To learn more about these types of problems, check out the following questions:
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