7. Use Structure Explain how you know the quotients $$ 540 \div 90 $$ and $$ 5,400 \div 900 $$ are equal without doing any computation.

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UpStudy AI · Accepted Answer

To determine why the quotients $$ 540 \div 90 $$ and $$ 5,\!400 \div 900 $$ are equal without performing any actual calculations, we can analyze the **structure** of the numbers involved, specifically by examining how the numerator and the denominator in each division problem are related through scaling.

### Step-by-Step Explanation:

1. **Identify the Relationship Between the Numerators:**
   - $$ 5,\!400 $$ is exactly **10 times** $$ 540 $$.
     $$
     5,\!400 = 10 \times 540
     $$

2. **Identify the Relationship Between the Denominators:**
   - $$ 900 $$ is exactly **10 times** $$ 90 $$.
     $$
     900 = 10 \times 90
     $$

3. **Express the Second Division in Terms of the First:**
   - Substitute the scaled values into the second division:
     $$
     \frac{5,\!400}{900} = \frac{10 \times 540}{10 \times 90}
     $$

4. **Simplify by Canceling the Common Factor:**
   - The factor of **10** in the numerator and the denominator cancels out:
     $$
     \frac{10 \times 540}{10 \times 90} = \frac{540}{90}
     $$

5. **Conclusion:**
   - After simplifying, both divisions reduce to the same expression:
     $$
     \frac{540}{90}
     $$
   - Therefore, 
     $$
     540 \div 90 = 5,\!400 \div 900
     $$

### Key Insight:

When both the numerator and the denominator of a fraction are **scaled by the same factor**, the overall value of the fraction **remains unchanged**. In this case, both the numerator and the denominator were multiplied by **10**, so the value of the quotient does not change.

This structural relationship ensures that:
$$
\frac{a \times k}{b \times k} = \frac{a}{b}
$$
where $$ k $$ is any non-zero constant. Applying this principle shows that $$ 540 \div 90 $$ and $$ 5,\!400 \div 900 $$ yield the same result without needing to perform the actual division.
Both $$ 540 \div 90 $$ and $$ 5,\!400 \div 900 $$ are equal because when you multiply both the numerator and the denominator by the same number (10 in this case), the value of the quotient remains the same.

Use Structure Explain how you know
the quotients and
are equal without doing any
computation.

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Use Structure Explain how you know the quotients and are equal without doing any computation.