The total age of two sisters is 108 years. One is 18 years older than the other. Find the ratio of the age of the older to that of the younger.
Answer (2)
The ratio of the older sister's age to the younger sister's age is 5 7 . This was found by establishing an equation based on their ages and solving for the younger sister's age. The older sister is 63 years old, while the younger sister is 45 years old.
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Let x represent the age of the younger sister, so the older sister's age is x + 18 .
Write the equation for the sum of their ages: x + ( x + 18 ) = 108 .
Solve for x : x = 45 , which means the younger sister is 45 years old and the older sister is 45 + 18 = 63 years old.
Calculate the ratio of the older sister's age to the younger sister's age and simplify: 45 63 = 5 7 . The final answer is 5 7 .
Explanation
Problem Analysis Let's analyze the problem. We know the total age of two sisters and the age difference between them. We need to find the ratio of the older sister's age to the younger sister's age.
Setting up the equation Let x be the age of the younger sister. Then the age of the older sister is x + 18 . The sum of their ages is x + ( x + 18 ) = 108 .
Solving for the younger sister's age Now, let's solve the equation for x :
x + ( x + 18 ) = 108
2 x + 18 = 108
2 x = 108 − 18
2 x = 90
x = 2 90
x = 45
So, the younger sister is 45 years old.
Finding the older sister's age The age of the older sister is x + 18 = 45 + 18 = 63 years.
Calculating the ratio Now, we need to find the ratio of the older sister's age to the younger sister's age:
Younger sister’s age Older sister’s age = 45 63
Simplifying the ratio To simplify the ratio, we can divide both the numerator and the denominator by their greatest common divisor, which is 9:
45 63 = 45 ÷ 9 63 ÷ 9 = 5 7
Final Answer Therefore, the ratio of the age of the older sister to the age of the younger sister is 5 7 .
Examples
Understanding ratios is very useful in everyday life. For example, if you are baking a cake and the recipe calls for a ratio of flour to sugar of 7:5, and you want to use 350 grams of flour, you can use this ratio to calculate how much sugar you need. In this case, you would need 250 grams of sugar. Ratios are also used in scaling recipes, calculating proportions, and understanding financial ratios.
