Ian's age is twice that of Emily now. 7 years ago, Ian was three times as old as Emily. Find their present ages.
Understand the Problem
The question asks to find the present ages of Ian and Emily based on the relationship of their ages now and 7 years ago. We will set up equations based on the information given and solve them to find their current ages.
Answer
Ian's age is $28$, and Emily's age is $14$.
Answer for screen readers
Ian's current age is ( 28 ), and Emily's current age is ( 14 ).
Steps to Solve
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Define Variables Let ( i ) represent Ian's current age and ( e ) represent Emily's current age.
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Set Up the Equations From the problem, we have two key relationships:
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Ian's age is twice Emily's age: $$ i = 2e $$
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Seven years ago, Ian was three times as old as Emily: $$ i - 7 = 3(e - 7) $$
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Substitute and Solve the Equations Substituting ( i = 2e ) into the second equation: $$ 2e - 7 = 3(e - 7) $$
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Expand and Simplify Expanding the right side: $$ 2e - 7 = 3e - 21 $$
Now, rearranging the equation: $$ 2e - 3e = -21 + 7 $$ $$ -e = -14 $$ $$ e = 14 $$
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Find Ian's Age Using ( e = 14 ) in the first equation: $$ i = 2e = 2(14) = 28 $$
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Summary of Age Calculation Thus, Ian's current age is ( 28 ) and Emily's current age is ( 14 ).
Ian's current age is ( 28 ), and Emily's current age is ( 14 ).
More Information
This problem illustrates how to set up and solve equations based on age relationships. It’s common in algebra to encounter problems where relationships change over time, making it crucial to translate verbal information into mathematical expressions.
Tips
- Not correctly translating the word problem into equations. Ensure that each relationship is accurately described.
- Forgetting to adjust the ages when considering past values (e.g., forgetting the subtraction of 7 years).
- Mismanaging signs during the algebraic manipulation, especially when separating variables.
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