To find the present ages of the two sisters, let's denote their current ages as follows:
- Let "y" represent the current age of the younger sister.
- Let "o" represent the current age of the older sister.
We need to solve the problem using the information given about their age ratios.
1. Given that the current ratio of their ages is 3:7, we can write the relationship as:
[tex]\[ \frac{y}{o} = \frac{3}{7} \][/tex]
2. In ten years, we know that the ratio of their ages will become 2:3. Therefore, we have:
[tex]\[ \frac{y + 10}{o + 10} = \frac{2}{3} \][/tex]
Now we have two equations:
[tex]\[ \frac{y}{o} = \frac{3}{7} \][/tex]
[tex]\[ \frac{y + 10}{o + 10} = \frac{2}{3} \][/tex]
From the first equation, we can express the age of the younger sister in terms of the older sister's age:
[tex]\[ y = \frac{3}{7}o \][/tex]
Next, substitute this expression into the second equation:
[tex]\[ \frac{\left( \frac{3}{7}o \right) + 10}{o + 10} = \frac{2}{3} \][/tex]
Simplify the numerator:
[tex]\[ \frac{\frac{3}{7}o + 10}{o + 10} = \frac{2}{3} \][/tex]
To eliminate the fraction, multiply both sides by the denominator [tex]\((o + 10)\)[/tex]:
[tex]\[ \frac{3}{7}o + 10 = \frac{2}{3}(o + 10) \][/tex]
Next, clear the fraction by multiplying every term by 21 (the least common multiple of 7 and 3):
[tex]\[ 21 \left( \frac{3}{7}o + 10 \right) = 21 \left( \frac{2}{3}(o + 10) \right) \][/tex]
This simplifies to:
[tex]\[ 3 \times 3o + 21 \times 10 = 7 \times 2(o + 10) \][/tex]
[tex]\[ 9o + 210 = 14(o + 10) \][/tex]
[tex]\[ 9o + 210 = 14o + 140 \][/tex]
Subtract [tex]\(9o\)[/tex] from both sides to solve for [tex]\(o\)[/tex]:
[tex]\[ 210 = 14o - 9o + 140 \][/tex]
[tex]\[ 210 = 5o + 140 \][/tex]
Subtract 140 from both sides:
[tex]\[ 70 = 5o \][/tex]
Divide both sides by 5:
[tex]\[ o = 14 \][/tex]
So, the current age of the older sister is 14 years.
Now, substitute [tex]\(o = 14\)[/tex] back into the expression for the younger sister's age:
[tex]\[ y = \frac{3}{7} \times 14 \][/tex]
[tex]\[ y = 6 \][/tex]
Therefore, the current ages of the sisters are:
- The younger sister is 6 years old.
- The older sister is 14 years old.
