Problem 44 - Prealgebra

Question

For an art project, Halle needs \(11 \frac{3}{8}\) inches of red ribbon and \(6 \frac{7}{9}\) inches of white ribbon. Which is the best estimate for the total amount of ribbon that she needs? A 8 in. B 10 in. C 18 in. D 26 in.

Step-by-Step Solution

Answer

Option C, 18 in., is the best estimate for the total ribbon.

Step 1: Understand the Problem

Halle needs a certain amount of red ribbon and a different amount of white ribbon. We need to estimate the total ribbon required by adding these two measurements together.

Step 2: Approximate the Fractions

First, approximate the fractional parts: \(\frac{3}{8} \approx \frac{1}{2}\) and \(\frac{7}{9} \approx 1\). This helps simplify the calculations for estimation.

Step 3: Estimate Each Mixed Number

Now, estimate the mixed numbers using the approximations: \(11 \frac{3}{8} \approx 11.5\) and \(6 \frac{7}{9} \approx 7\). These are easier to add up.

Step 4: Sum the Estimated Numbers

Add the estimated numbers together: \(11.5 + 7 = 18.5\). Round this down to the nearest whole number for an estimate, which is 18.

Step 5: Choose the Best Answer

Compare the estimated sum with the options given: 8 in., 10 in., 18 in., and 26 in. The closest estimate to 18.5 is 18 in.

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are used to express quantities that are greater than whole numbers but not quite at the next whole number level. For example, in the exercise, Halle needs 11\(\frac{3}{8}\) inches of red ribbon. Here, 11 is the whole number, and \(\frac{3}{8}\) is the fraction.
When estimating sums involving mixed numbers, it's often helpful to separately consider the whole numbers and the fractional parts. This approach simplifies calculations and leads to quicker approximations. In the exercise, both the red and white ribbon measurements involved mixed numbers. By focusing on the whole numbers first, you can quickly gain a rough idea of the total before fine-tuning with the fractional parts.
Knowing how to handle mixed numbers effectively can improve your arithmetic skills and make estimating easier and more accurate.

Working with Fractions

Fractions represent parts of a whole, and often appear in scenarios like measuring, cooking, and art projects like Halle's.
In the problem, Halle needs to estimate the total ribbon needed, which involves working with the fractions \(\frac{3}{8}\) and \(\frac{7}{9}\). To simplify estimation, we often approximate fractions to the nearest simple fractions or whole numbers. For example, \(\frac{3}{8}\) is approximately \(\frac{1}{2}\) because it's close to half, and \(\frac{7}{9}\) is nearly 1 because it is only two parts away from a complete whole.
Such approximations can make addition easier, especially when quick, rough calculations are all that are needed. While exact values are important in precise situations, estimations like these are often enough for decision-making in projects.

The Importance of Addition in Estimation

Addition is the process of finding the total or sum by combining numbers. When estimating sums, addition becomes a crucial skill to ensure an accurate understanding of the total needed.
In Halle's art project, she adds the estimated lengths of the red and white ribbons to find the total length needed. By approximating \(11 \frac{3}{8}\) to 11.5 and \(6 \frac{7}{9}\) to 7, the addition becomes straightforward: 11.5 + 7 = 18.5.
Often, after the arithmetic is done, you may need to round the result to the nearest whole number for simplicity. Here, 18.5 is rounded down to 18. Comparing this to the provided options gives the best answer of 18 inches.
Reliable addition skills make it simpler to estimate sums and choose answers confidently, especially when the task involves mixed numbers and fractions.