Answer :
Sure! Let's break down the problem to find the unit rate equivalent to the given ratios.
We are dealing with the following ratio:
- Flour for every [tex]\( \frac{1}{2} \)[/tex] teaspoon of baking soda and every [tex]\( \frac{1}{4} \)[/tex] teaspoon of salt.
First, let's find the unit rate of salt per teaspoon of baking soda.
### Step 1: Calculate the current ratio for salt and baking soda
We have:
- [tex]\( \frac{1}{4} \)[/tex] teaspoon of salt per [tex]\( \frac{1}{2} \)[/tex] teaspoon of baking soda
To find the amount of salt per 1 teaspoon of baking soda, divide the salt amount by the baking soda amount:
[tex]\[ \text{Salt per teaspoon of baking soda} = \frac{\frac{1}{4}}{\frac{1}{2}} \][/tex]
[tex]\[ = \frac{1/4}{1/2} = \frac{1}{4} \times \frac{2}{1} = \frac{1 \times 2}{4 \times 1} = \frac{2}{4} = \frac{1}{2} \][/tex]
So, 1 teaspoon of baking soda corresponds to [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt.
### Step 2: Compare with given rates to check for equivalence
#### Compare with each given option:
1. [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt per 1 teaspoon of baking soda: This matches our calculated ratio directly as both are [tex]\( \frac{1}{2} \)[/tex].
2. 2 teaspoons of salt per 1 cup of flour: This doesn't immediately relate to the salt per baking soda rate we calculated.
3. 2 teaspoons of salt per 1 teaspoon of baking soda: This ratio is 2, which does not match our calculated ratio of [tex]\( \frac{1}{2} \)[/tex].
4. 1 teaspoon of baking soda per 2 teaspoons of salt: To check if this matches, we can convert it back to salt per baking soda:
If 1 teaspoon of baking soda is for 2 teaspoons of salt, then:
[tex]\[ \text{Salt per teaspoon of baking soda} = \frac{1}{2} \text{ per teaspoon of salt} = \frac{1}{2} \][/tex]
This also gives a ratio of [tex]\( \frac{1}{2} \)[/tex], which matches our calculated ratio.
### Conclusion:
- The unit rates equivalent to the ratio of [tex]\( \frac{1}{2} \)[/tex] teaspoon of baking soda per [tex]\( \frac{1}{4} \)[/tex] teaspoon of salt are:
1. [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt per 1 teaspoon of baking soda
2. 1 teaspoon of baking soda per 2 teaspoons of salt
So, the correct answer is:
- [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt per 1 teaspoon of baking soda
- 1 teaspoon of baking soda per 2 teaspoons of salt
We are dealing with the following ratio:
- Flour for every [tex]\( \frac{1}{2} \)[/tex] teaspoon of baking soda and every [tex]\( \frac{1}{4} \)[/tex] teaspoon of salt.
First, let's find the unit rate of salt per teaspoon of baking soda.
### Step 1: Calculate the current ratio for salt and baking soda
We have:
- [tex]\( \frac{1}{4} \)[/tex] teaspoon of salt per [tex]\( \frac{1}{2} \)[/tex] teaspoon of baking soda
To find the amount of salt per 1 teaspoon of baking soda, divide the salt amount by the baking soda amount:
[tex]\[ \text{Salt per teaspoon of baking soda} = \frac{\frac{1}{4}}{\frac{1}{2}} \][/tex]
[tex]\[ = \frac{1/4}{1/2} = \frac{1}{4} \times \frac{2}{1} = \frac{1 \times 2}{4 \times 1} = \frac{2}{4} = \frac{1}{2} \][/tex]
So, 1 teaspoon of baking soda corresponds to [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt.
### Step 2: Compare with given rates to check for equivalence
#### Compare with each given option:
1. [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt per 1 teaspoon of baking soda: This matches our calculated ratio directly as both are [tex]\( \frac{1}{2} \)[/tex].
2. 2 teaspoons of salt per 1 cup of flour: This doesn't immediately relate to the salt per baking soda rate we calculated.
3. 2 teaspoons of salt per 1 teaspoon of baking soda: This ratio is 2, which does not match our calculated ratio of [tex]\( \frac{1}{2} \)[/tex].
4. 1 teaspoon of baking soda per 2 teaspoons of salt: To check if this matches, we can convert it back to salt per baking soda:
If 1 teaspoon of baking soda is for 2 teaspoons of salt, then:
[tex]\[ \text{Salt per teaspoon of baking soda} = \frac{1}{2} \text{ per teaspoon of salt} = \frac{1}{2} \][/tex]
This also gives a ratio of [tex]\( \frac{1}{2} \)[/tex], which matches our calculated ratio.
### Conclusion:
- The unit rates equivalent to the ratio of [tex]\( \frac{1}{2} \)[/tex] teaspoon of baking soda per [tex]\( \frac{1}{4} \)[/tex] teaspoon of salt are:
1. [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt per 1 teaspoon of baking soda
2. 1 teaspoon of baking soda per 2 teaspoons of salt
So, the correct answer is:
- [tex]\( \frac{1}{2} \)[/tex] teaspoon of salt per 1 teaspoon of baking soda
- 1 teaspoon of baking soda per 2 teaspoons of salt
