Sure, let's solve this problem step-by-step.
1. Identify the given values:
- Amira has [tex]\(\frac{3}{4}\)[/tex] of a bag of cat food.
- The cat eats [tex]\(\frac{1}{10}\)[/tex] of a bag each week.
2. Understand the question:
- We need to find out how many weeks the food will last.
3. Set up the problem:
- We start with [tex]\(\frac{3}{4}\)[/tex] of a bag.
- Each week, the cat consumes [tex]\(\frac{1}{10}\)[/tex] of a bag.
4. Calculate the number of weeks the food will last:
- To find how many weeks the food will last, we need to divide the total amount of food by the weekly consumption.
- This can be represented mathematically as:
[tex]\[
\text{Number of weeks} = \frac{\text{Total amount of food}}{\text{Weekly consumption}}
\][/tex]
5. Substitute the given values into the formula:
- Total amount of food: [tex]\(\frac{3}{4}\)[/tex] of a bag.
- Weekly consumption: [tex]\(\frac{1}{10}\)[/tex] of a bag.
[tex]\[
\text{Number of weeks} = \frac{\frac{3}{4}}{\frac{1}{10}}
\][/tex]
6. Performing the division:
- To divide by a fraction, multiply by its reciprocal. Thus,
[tex]\[
\frac{\frac{3}{4}}{\frac{1}{10}} = \frac{3}{4} \times \frac{10}{1} = \frac{3 \times 10}{4 \times 1} = \frac{30}{4}
\][/tex]
7. Simplify the fraction:
- [tex]\(\frac{30}{4}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{30}{4} = \frac{30 \div 2}{4 \div 2} = \frac{15}{2}
\][/tex]
- [tex]\(\frac{15}{2}\)[/tex] is equal to 7.5 when expressed as a decimal.
8. Conclusion:
- The food will last for 7.5 weeks.
Thus, Amira's cat food will last for 7.5 weeks.
