Question

7.3.ip-8
of 54 cities, 7 have a lake and 23 have a river running through them. what fraction of these 54 cities have
a lake or a river running through them? show your work.

which expression can be used to find the fraction of cities that have a lake or a river running
through them? select the correct choice and fill in the answer box to complete your choice.
(type a whole number or a fraction.)

a. \\(\frac{23}{54} - \frac{7}{54} = \frac{\square}{54} \div \frac{6}{6} = \square\\)

b. \\(\frac{23}{54} - \frac{7}{54} = \frac{\square}{54} \div \frac{54}{54} = \square\\)

c. \\(\frac{7}{54} + \frac{23}{54} = \frac{\square}{54} \div \frac{54}{54} = \square\\)

d. \\(\frac{7}{54} + \frac{23}{54} = \frac{\square}{54} \div \frac{6}{6} = \square\\)

Explanation:

Step1: Understand the Problem

We need to find the fraction of cities with a lake or a river. The number of cities with a lake is 7, with a river is 23, total cities 54. So we use the principle of inclusion - exclusion for sets (here, since there's no overlap mentioned, we assume no cities have both, so we add the fractions). The fraction with lake is $\frac{7}{54}$, with river is $\frac{23}{54}$. So we add them: $\frac{7}{54}+\frac{23}{54}$. Then, when dividing by a fraction, we multiply by its reciprocal. But let's check the options.

Step2: Analyze Each Option

- Option A: Subtraction, which is wrong as we need to find "or", so addition. Eliminate A.
- Option B: Subtraction, wrong. Eliminate B.
- Option C: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}\div1 = \frac{30}{54}=\frac{5}{9}$. Wait, but let's check the operation. Wait, $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$. But $\frac{54}{54}=1$, so $\frac{30}{54}\div1=\frac{30}{54}$. But also, let's check the other part. Wait, maybe the problem has a typo, but let's see the addition. The correct expression for "or" (no overlap) is adding the two fractions. Then, when we have $\frac{7 + 23}{54}=\frac{30}{54}$, and then dividing by $\frac{54}{54}$ is just $\frac{30}{54}$. But also, let's check Option D. Wait, Option D has $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{6}{6}$? Wait, no, Option D is $\frac{30}{54}\div\frac{6}{6}$? Wait, no, Option D is $\frac{30}{54}\div\frac{6}{6}$? Wait, the original Option D is $\frac{7}{54}+\frac{23}{54}=\frac{\square}{54}\div\frac{6}{6}$. Wait, $\frac{6}{6}=1$, so $\frac{30}{54}\div1=\frac{30}{54}$. But wait, maybe the problem has a mistake, but let's re - examine. Wait, the key is that to find the fraction of cities with lake or river, we add the fractions of cities with lake and with river. So the first part is $\frac{7}{54}+\frac{23}{54}$. Now, looking at the options, Option C: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}$. Option D: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{6}{6}=\frac{30}{54}$. Wait, but maybe the problem intended that after adding, we simplify. Wait, $\frac{7 + 23}{54}=\frac{30}{54}=\frac{5}{9}$. Now, let's check the operations in the options. Option C: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}\div1=\frac{30}{54}=\frac{5}{9}$. Option D: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{6}{6}=\frac{30}{54}\div1=\frac{30}{54}$. But the difference is in the divisor. Wait, maybe there's a mistake in the problem, but the correct approach is to add the two fractions. So the expression should be $\frac{7}{54}+\frac{23}{54}$, then simplify. Now, looking at the options, Option C has $\frac{7}{54}+\frac{23}{54}=\frac{\square}{54}\div\frac{54}{54}$, which is correct because $\frac{54}{54}=1$, so dividing by 1 doesn't change the value. Option D has $\frac{6}{6}$, which is also 1, but let's check the numerator. $\frac{7 + 23}{54}=\frac{30}{54}$, so the square is 30. Then, for Option C: $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}=\frac{5}{9}$. For Option D: $\frac{30}{54}\div\frac{6}{6}=\frac{30}{54}=\frac{5}{9}$. Wait, but maybe the problem has a typo, but the key is that the correct operation is addition. So the correct option is C (or D? Wait, no, let's re - check the options. Wait, Option C: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}$. Option D: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{6}{6}=\frac{30}{54}$. But the difference is in the divisor. Wait, maybe the problem meant to have a common factor. Wait, 7 and 23: do they have a common factor? 7 is prime, 23 is prime, so no. Wait, 30 and 54 have a common factor of 6. So $\frac{30}{54}=\frac{30\div6}{54\div6}=\frac{5}{9}$. Ah! So when we divide $\frac{30}{54}$ by $\frac{6}{6}$, no, wait, $\frac{30}{54}\div\frac{6}{6}$ is same as $\frac{30}{54}$. But if we divide $\frac{30}{54}$ by $\frac{6}{54}$? No, the options have $\frac{54}{54}$ or $\frac{6}{6}$. Wait, maybe the problem has a mistake, but the correct expression for the fraction is $\frac{7 + 23}{54}=\frac{30}{54}=\frac{5}{9}$. And the operation in Option C: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}$, and simplifying $\frac{30}{54}$ by dividing numerator and denominator by 6 gives $\frac{5}{9}$. Wait, but the option's operation is $\frac{30}{54}\div\frac{54}{54}$, which is $\frac{30}{54}$, then maybe the next step is to simplify, but the question is about the expression to find the fraction. So the correct expression is adding the two fractions, so Option C or D? Wait, no, $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}$, and $\frac{30}{54}\div\frac{6}{6}$ is also $\frac{30}{54}$. But let's check the arithmetic. $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$. Now, $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}\times\frac{54}{54}=\frac{30}{54}$ (since $\frac{54}{54}=1$). And $\frac{30}{54}\div\frac{6}{6}=\frac{30}{54}\times\frac{6}{6}=\frac{30}{54}$. But the key is that the first part is addition. So the correct option is C (because we add the two fractions, then divide by 1 (since $\frac{54}{54}=1$), which is the same as the fraction itself, and then we can simplify. Wait, but let's check the numbers. 7 + 23 = 30, so the numerator is 30. Then, for Option C: $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}$, then $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}$, and simplifying $\frac{30}{54}$ by dividing numerator and denominator by 6 gives $\frac{5}{9}$. So the correct option is C, with the first square as 30, and the final answer as $\frac{5}{9}$. But let's confirm the operation. When we add $\frac{7}{54}$ and $\frac{23}{54}$, we get $\frac{30}{54}$. Then, dividing by $\frac{54}{54}$ (which is 1) leaves us with $\frac{30}{54}$, which simplifies to $\frac{5}{9}$. So the correct option is C, with the first box as 30 and the final answer as $\frac{5}{9}$.

Step3: Calculate the Values

First, add the fractions: $\frac{7}{54}+\frac{23}{54}=\frac{7 + 23}{54}=\frac{30}{54}$. Then, $\frac{30}{54}\div\frac{54}{54}=\frac{30}{54}\div1=\frac{30}{54}=\frac{5}{9}$. So the first square is 30, and the final answer is $\frac{5}{9}$.

Answer:

The correct option is C. $\frac{7}{54}+\frac{23}{54}=\frac{30}{54}\div\frac{54}{54}=\frac{5}{9}$ (the first box is 30, the final answer is $\frac{5}{9}$)