Adaptive Practice Without multiplying, decide which symbol to use: $$ ,

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Adaptive Practice Without multiplying, decide which symbol to use: $$ >,< $$, or $$ = $$. Explain how you decided. $$ 4 \frac{1}{7} \times \frac{1}{5}-4 \frac{1}{4} $$

UpStudy AI · Accepted Answer

To compare the two expressions $$ 4 \frac{1}{7} \times \frac{1}{5} $$ and $$ 4 \frac{1}{4} $$ without multiplying, we can first convert the mixed numbers into improper fractions.

1. **Convert $$ 4 \frac{1}{7} $$ to an improper fraction:**
   $$
   4 \frac{1}{7} = 4 + \frac{1}{7} = \frac{4 \times 7 + 1}{7} = \frac{28 + 1}{7} = \frac{29}{7}
   $$

2. **Convert $$ 4 \frac{1}{4} $$ to an improper fraction:**
   $$
   4 \frac{1}{4} = 4 + \frac{1}{4} = \frac{4 \times 4 + 1}{4} = \frac{16 + 1}{4} = \frac{17}{4}
   $$

Now we have the two expressions as:
- $$ 4 \frac{1}{7} \times \frac{1}{5} = \frac{29}{7} \times \frac{1}{5} = \frac{29}{35} $$
- $$ 4 \frac{1}{4} = \frac{17}{4} $$

3. **Now we need to compare $$ \frac{29}{35} $$ and $$ \frac{17}{4} $$.** To do this, we can find a common denominator. The least common multiple of 35 and 4 is 140.

4. **Convert both fractions to have a denominator of 140:**
   - For $$ \frac{29}{35} $$:
   $$
   \frac{29}{35} = \frac{29 \times 4}{35 \times 4} = \frac{116}{140}
   $$
   - For $$ \frac{17}{4} $$:
   $$
   \frac{17}{4} = \frac{17 \times 35}{4 \times 35} = \frac{595}{140}
   $$

5. **Now we can compare $$ \frac{116}{140} $$ and $$ \frac{595}{140} $$:**
   Since $$ 116 < 595 $$, we have:
   $$
   \frac{29}{35} < \frac{17}{4}
   $$

Thus, the final comparison is:
$$
4 \frac{1}{7} \times \frac{1}{5} < 4 \frac{1}{4}
$$

Therefore, the symbol to use is $$ < $$.
$$ < $$

Adaptive Practice Without multiplying, decide which symbol to use: \( >,< \), or \( = \). Explain how you decided. \[ 4 \frac{1}{7} \times \frac{1}{5}-4 \frac{1}{4} \]

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