Problem 55 - Algebra 1

Question

Suppose you are 14 years old and your brother is 4 years old. a. In \(t\) years, your age will be \(14+t\). What will your brother's age be? b. Write the ratio of your age in \(t\) years to your brother's age in \(t\) years. Then use long division to rewrite this ratio. c. Use the rewritten ratio to find the ratio of your ages now, in 5 years, in 10 years, in 25 years, in 50 years, and in 80 years. d. Use your answers to part (c). Is the ratio of your ages getting smaller or larger as time goes by? What value do the ratios approach? e. Writing Look at the original form of the ratio and the rewritten form of the ratio. Which form of the ratio makes it easier for you to recognize the trend that you described in part (d)? Explain your choice.

Step-by-Step Solution

Answer

a. Brother's age in \(t\) years will be \(4+t\). b. The ratio is \(\frac{14+t}{4+t}\) which can be rewritten as \(1+\frac{10}{4+t}\) c. The ratios of their ages in 0, 5, 10, 25, 50, 80 years are respectively 3.5, 1.9, 1.67, 1.36, 1.18 and 1.1. d. The ratio is getting smaller as time goes by, approaching a value of 1. e. The rewritten form of the ratio makes it easier to recognize the trend as compared to the original ratio form.

Step 1: Determine Brother's Age in T Years

If the elder sibling's age is given by the equation \(14 + t\), where \(t\) is the number of additional years from now, the younger sibling's age in \(t\) years would be \(4 + t\). It is determined by adding \(t\) to the younger sibling's current age which is 4.

Step 2: Constructing the Age Ratio

The ratio of the elder sibling's age to the younger sibling's age in \(t\) years is \(\frac{14 + t}{4 + t}\). This ratio represents the relationship between the two ages in \(t\) years

Step 3: Simplify the Ratio with Long Division

By using long division method, the ratio can be rewritten as \(1 + \frac{10}{4 + t}\)

Step 4: Calculation of Ratios at Various Time Points

Now, substituting the values of \(t\) as 0, 5, 10, 25, 50 and 80 years into the rewritten ratio will give the ratios of their ages at each of these times.

Step 5: Analysis of Ratio Trend

In all the calculated ratios, as \(t\) becomes large, the ratio approaches the constant value of 1 since the fraction \(\frac{10}{4 + t}\) tends towards 0 with an increase in \(t\). This implies that, as time goes by, the difference in age becomes less significant.

Step 6: Compare Ratio Forms for Trend Recognition

Evaluating which form of the ratio makes it easier to predict the trend, it can be reasoned that with the rewritten form of the ratio, \(1 + \frac{10}{4 + t}\), it's straightforward to see that as \(t\) increases, the ratio approaches a value of 1, while in the original ratio form, the trend isn't immediately obvious.

Key Concepts

Simplifying Ratios

When it comes to understanding relationships between numbers, ratios are a fundamental tool. Simplifying ratios can make them easier to work with and to interpret. In algebra, ratios are often used to compare two quantities, like the ages of two siblings. For example, if one sibling is 14 years old and the other is 4 years old, their age ratio can be expressed as \(\frac{14}{4}\). To simplify, you divide both the numerator and the denominator by their greatest common divisor, which in this case is 2, resulting in \(\frac{7}{2}\), a simpler ratio to grasp.

Simplifying becomes even more useful when variables are involved. Take the dynamic ratio \(\frac{14 + t}{4 + t}\) that changes with time \(t\). By using basic algebraic skills like long division, it’s possible to break down the ratio into a more digestible form, \(1 + \frac{10}{4 + t}\). This new form reveals that no matter what the value of \(t\) is, there’s always a base value of 1 in the ratio, indicating an inherent baseline comparison between the siblings' ages.

Long Division in Algebra

Long division isn’t just for arithmetic; it can also be a powerful tool in algebra for simplifying expressions, including ratios. When ratios involve variables, long division can help reveal underlying trends. Consider the age ratio \(\frac{14 + t}{4 + t}\). By applying long division, you get \(1 + \frac{10}{4 + t}\).

Why Use Long Division?

Using long division, we can simplify the ratio to its essence, making it easier to digest and understand. When the age ratio is in its original form, identifying trends as \(t\) changes can be complex. However, long division clarifies that as the value of \(t\) increases, the ratio converges towards 1, because the \(\frac{10}{4 + t}\) term becomes negligible. This insight is crucial in comprehending age difference trends over time.

Age Difference Trends

Examining age difference trends can provide fascinating insights into how relationships change over time. In our sibling example, the age ratio \(1 + \frac{10}{4 + t}\) hints at an important trend: as time passes, the significance of the age difference diminishes. This is because as \(t\) grows larger, the fractional part \(\frac{10}{4 + t}\) approaches 0 and the entire ratio trends towards 1.

Effect of Time on Age Differences

In the case of our siblings, whether the time \(t\) is 5 years or 50 years in the future, the denominator in \(\frac{10}{4 + t}\) becomes substantially larger, lessening the impact of the numerator, 10. This means that while their actual age difference remains the same, it becomes proportionally less significant compared to their total age. In essence, time acts as a great equalizer in terms of age differences, a concept that has applications in understanding social dynamics, inheritance patterns, and even population studies.