The ratio of Amber's age to her brother Zach's age is $1: 2$. Find the missing age.
The problem provides the ratio of Amber's age to her brother Zach's age, which is \(1:2\). This ratio implies that for every year of Amber's age, Zach is twice as old. We can express this relationship mathematically as \(\frac{A}{Z} = \frac{1}{2}\), where \(A\) represents Amber's age and \(Z\) represents Zach's age. This can also be rewritten as \(Z = 2A\) or \(A = \frac{Z}{2}\). We will use these relationships to find the missing ages in the two given scenarios.
For the first scenario, we are given that Amber is 10 years old (\(A = 10\)). To find Zach's age (\(Z\)), we use the relationship \(Z = 2A\):
\[Z = 2 \times 10 = 20.\]
Thus, when Amber is 10, Zach is 20 years old.
For the second scenario, we are given that Zach is 26 years old (\(Z = 26\)). To find Amber's age (\(A\)), we use the relationship \(A = \frac{Z}{2}\):
\[A = \frac{26}{2} = 13.\]
Thus, when Zach is 26, Amber is 13 years old.
Ratios like \(1:2\) are essential in everyday life, helping us understand proportional relationships. For example, if you are following a recipe that requires 1 cup of sugar for every 2 cups of flour, you are using a \(1:2\) ratio. If you want to make a larger batch using 3 cups of sugar, you would need to use \(3 \times 2 = 6\) cups of flour. Mastering these calculations helps ensure that mixtures, designs, and financial plans remain consistent and accurate.