Verified Answer :

  • Analyze \(\frac{7}{8}\) as being close to \(1\).
  • Analyze \(\frac{2}{5}\) as being close to \(\frac{1}{2}\).
  • Calculate the estimate: \(1 - \frac{1}{2} = \frac{1}{2}\).
  • Verify the exact value \(\frac{19}{40} = 0.475\), confirming it is closest to \(\boxed{\frac{1}{2}}\).

Explanation

  1. Estimating the Fractions

To estimate the value of the expression \(\frac{7}{8} - \frac{2}{5}\), we can compare each fraction to standard benchmarks like \(0\), \(\frac{1}{2}\), and \(1\).

  • The fraction \(\frac{7}{8}\) has a numerator very close to its denominator, so it is just slightly less than \(1\).
  • The fraction \(\frac{2}{5}\) can be written as \(0.4\). Since \(\frac{1}{2} = 0.5\), \(\frac{2}{5}\) is quite close to \(\frac{1}{2}\).

By using these approximations, our estimate for the expression is \(1 - \frac{1}{2} = \frac{1}{2}\).

  1. Calculating the Exact Value

To verify how close our estimate is, let's calculate the exact value of the expression \(\frac{7}{8} - \frac{2}{5}\). We first find a common denominator, which is \(8 \times 5 = 40\).

  • The expression becomes \(\frac{7 \times 5}{8 \times 5} - \frac{2 \times 8}{5 \times 8} = \frac{35}{40} - \frac{16}{40} = \frac{19}{40}\).

Converting this to a decimal, \(\frac{19}{40} = 0.475\).

  1. Comparing with Options

Now we compare the exact value \(0.475\) with the provided options:

  • \(0\)
  • \(\frac{1}{2} = 0.5\)
  • \(1\)
  • \(1 \frac{1}{2} = 1.5\)

The absolute differences are:

  • \(|0.475 - 0| = 0.475\)
  • \(|0.475 - 0.5| = 0.025\)
  • \(|0.475 - 1| = 0.525\)
  • \(|0.475 - 1.5| = 1.025\)

Since \(0.025\) is the smallest difference, the value \(\frac{1}{2}\) is the closest estimate.

Examples

Estimating fractions is very useful in everyday life, such as when cooking or shopping. For instance, if a recipe calls for \(\frac{7}{8}\) cup of flour but you already used \(\frac{2}{5}\) of your measuring cup for another ingredient, knowing that you have roughly \(\frac{1}{2}\) cup remaining helps you quickly determine if you have enough supplies for the next step without needing to perform exact calculations. Understanding these approximations helps make quick decisions when precision is not immediately required.

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