Math Problem Statement

Use the number line above to correctly compare the following numbers: π, √7.29, 16/7.

Solution

The problem asks you to compare three numbers: π\pi, 7.29\sqrt{7.29}, and 167\frac{16}{7}, using the given number line ranging from 0 to 4.

Step 1: Calculate the approximate values of the numbers

  • π3.1416\pi \approx 3.1416
  • 7.29=2.7\sqrt{7.29} = 2.7 (since 2.72=7.292.7^2 = 7.29)
  • 1672.2857\frac{16}{7} \approx 2.2857 (since 16÷72.285716 \div 7 \approx 2.2857)

Step 2: Place these values on the number line

  • π3.1416\pi \approx 3.1416 is slightly above 3.
  • 7.29=2.7\sqrt{7.29} = 2.7 is located between 2 and 3, closer to 3.
  • 1672.2857\frac{16}{7} \approx 2.2857 is located between 2 and 3, but closer to 2.

Step 3: Compare the numbers

We can now compare their positions:

  • 1672.2857\frac{16}{7} \approx 2.2857 is the smallest.
  • Next is 7.29=2.7\sqrt{7.29} = 2.7.
  • Finally, π3.1416\pi \approx 3.1416 is the largest.

Step 4: Analyze the answer choices

  • A. π<7.29\pi < \sqrt{7.29} — Incorrect, because π>7.29\pi > \sqrt{7.29}.
  • B. 167<π\frac{16}{7} < \pi — Correct, because 1672.2857\frac{16}{7} \approx 2.2857 and π3.1416\pi \approx 3.1416.
  • C. π<167\pi < \frac{16}{7} — Incorrect, because π>167\pi > \frac{16}{7}.
  • D. 7.29<167\sqrt{7.29} < \frac{16}{7} — Incorrect, because 7.29>167\sqrt{7.29} > \frac{16}{7}.

Correct Answer: B

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you approximate square roots without a calculator?
  2. Why is π\pi an irrational number, and how is it represented on a number line?
  3. Can 167\frac{16}{7} be expressed as a mixed fraction, and what would it be?
  4. How does the number line help in comparing decimal approximations?
  5. What are the common ways to estimate fractions like 167\frac{16}{7}?

Tip: When comparing fractions, convert them to decimals to easily position them on a number line.

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Math Problem Analysis

Mathematical Concepts

Number Line
Decimal Approximation
Square Roots
Fractions
Irrational Numbers

Formulas

π ≈ 3.1416
√7.29 = 2.7
16/7 ≈ 2.2857

Theorems

Square Root Calculation
Fraction to Decimal Conversion

Suitable Grade Level

Grades 6-8