Answer :
To determine which equations are equivalent to the division equation [tex]\(\frac{1}{10} \div 7=?\)[/tex], we can rewrite the division as multiplication involving the unknown, “?”.
### Step-by-step solution:
1. Initial Division Equation:
[tex]\[
\frac{1}{10} \div 7 = ?
\][/tex]
This is equivalent to finding a number that, when multiplied by 7, gives [tex]\(\frac{1}{10}\)[/tex].
2. Rewriting as a Multiplication:
To convert the division into a multiplication, we use the property that dividing by a number is equivalent to multiplying by its reciprocal. Thus:
[tex]\[
\frac{1}{10} \div 7 = \text{means} \; \frac{1}{10} \times \frac{1}{7} = ?
\][/tex]
3. Equivalent Equations:
- We need to identify the equivalent equations from the given options:
- [tex]\(7 \times ? = \frac{1}{10}\)[/tex]
- [tex]\(? \times 7 = \frac{1}{10}\)[/tex]
- [tex]\(\frac{1}{10} \times ? = 7\)[/tex]
- [tex]\(? \times \frac{1}{10} = 7\)[/tex]
4. Checking Each Equation:
- Equation 1: [tex]\(7 \times ? = \frac{1}{10}\)[/tex]
- This is correct because solving for "?" gives [tex]\(\frac{1}{10} \div 7\)[/tex].
- Equation 2: [tex]\(? \times 7 = \frac{1}{10}\)[/tex]
- This is also correct as it mirrors equation 1.
- Equation 3: [tex]\(\frac{1}{10} \times ? = 7\)[/tex]
- Incorrect because it suggests that multiplying by "?" results in a number much larger than possible (7), which is different from what [tex]\(\frac{1}{10} \div 7\)[/tex] implies.
- Equation 4: [tex]\(? \times \frac{1}{10} = 7\)[/tex]
- Incorrect because it means "?" would be 70, which is not the result of [tex]\(\frac{1}{10} \div 7\)[/tex].
5. Solution:
- The correct equivalent equations are:
- [tex]\(7 \times ? = \frac{1}{10}\)[/tex]
- [tex]\(? \times 7 = \frac{1}{10}\)[/tex]
These steps confirm that both "7 times something equals [tex]\(\frac{1}{10}\)[/tex]" and "that something times 7 equals [tex]\(\frac{1}{10}\)[/tex]" represent the same idea as [tex]\(\frac{1}{10} \div 7\)[/tex].
### Step-by-step solution:
1. Initial Division Equation:
[tex]\[
\frac{1}{10} \div 7 = ?
\][/tex]
This is equivalent to finding a number that, when multiplied by 7, gives [tex]\(\frac{1}{10}\)[/tex].
2. Rewriting as a Multiplication:
To convert the division into a multiplication, we use the property that dividing by a number is equivalent to multiplying by its reciprocal. Thus:
[tex]\[
\frac{1}{10} \div 7 = \text{means} \; \frac{1}{10} \times \frac{1}{7} = ?
\][/tex]
3. Equivalent Equations:
- We need to identify the equivalent equations from the given options:
- [tex]\(7 \times ? = \frac{1}{10}\)[/tex]
- [tex]\(? \times 7 = \frac{1}{10}\)[/tex]
- [tex]\(\frac{1}{10} \times ? = 7\)[/tex]
- [tex]\(? \times \frac{1}{10} = 7\)[/tex]
4. Checking Each Equation:
- Equation 1: [tex]\(7 \times ? = \frac{1}{10}\)[/tex]
- This is correct because solving for "?" gives [tex]\(\frac{1}{10} \div 7\)[/tex].
- Equation 2: [tex]\(? \times 7 = \frac{1}{10}\)[/tex]
- This is also correct as it mirrors equation 1.
- Equation 3: [tex]\(\frac{1}{10} \times ? = 7\)[/tex]
- Incorrect because it suggests that multiplying by "?" results in a number much larger than possible (7), which is different from what [tex]\(\frac{1}{10} \div 7\)[/tex] implies.
- Equation 4: [tex]\(? \times \frac{1}{10} = 7\)[/tex]
- Incorrect because it means "?" would be 70, which is not the result of [tex]\(\frac{1}{10} \div 7\)[/tex].
5. Solution:
- The correct equivalent equations are:
- [tex]\(7 \times ? = \frac{1}{10}\)[/tex]
- [tex]\(? \times 7 = \frac{1}{10}\)[/tex]
These steps confirm that both "7 times something equals [tex]\(\frac{1}{10}\)[/tex]" and "that something times 7 equals [tex]\(\frac{1}{10}\)[/tex]" represent the same idea as [tex]\(\frac{1}{10} \div 7\)[/tex].
