Question

there are $97$ students in south high school. south high school offers only chinese and spanish. there are $8$ more students in chinese than in spanish, and every student takes at least one language. if $26$ students take only spanish, then how many take both languages?

Answer 1

Substitute C = S + 8 into the total student equation, simplify, and solve. With S = 63, substitute into B = S - 26 to find B = 37. Hence, 37 students take both languages.

Let's go through the problem step by step:

Given Information:

- Total number of students: C + S - B = 97 (Equation 1)

- There are 8 more students in Chinese than in Spanish: C = S + 8 (Equation 2)

- 26 students take only Spanish: S - B = 26 (Equation 3)

Step 1: Substitute C from Equation 2 into Equation 1:

(S + 8) + S - B = 97

Step 2: Simplify and Rearrange:

2S + 8 - B = 97

Step 3: Use Equation 3 to Express B in Terms of S :

B = S - 26

Step 4: Substitute B into the Modified Equation 1:

2S + 8 - (S - 26) = 97

Step 5: Simplify and Solve for S :

2S + 8 - S + 26 = 97

S + 34 = 97

S = 63

Step 6: Find B using the value of S from Equation 3:

B = S - 26

B = 63 - 26

B = 37

Therefore, according to this analysis, 37 students take both languages.

Answer 2

37 students take both languages.

To solve this problem, let's use algebra and step through it logically. We'll use these variables:

- Let C be the number of students taking Chinese.

- Let S be the number of students taking Spanish.

- Let B be the number of students taking both languages.

From the problem, we know:

1. Total number of students: C + S - B = 97 (subtract B to avoid double-counting the students who take both languages).

2. There are 8 more students in Chinese than in Spanish: C = S + 8 .

3. 26 students take only Spanish: S - B = 26 (since S includes students who take both).

Now, let's solve these equations step by step.

Step 1: Substitute C from Equation 2 into Equation 1

Since C = S + 8 , we can replace C in Equation 1:

S + 8 + S - B = 97

2S + 8 - B = 97

Step 2: Solve for B using Equations 3 and the modified Equation 1**

From Equation 3, B = S - 26 Substitute this into the modified Equation 1:

2S + 8 - (S - 26) = 97

2S + 8 - S + 26 = 97

S + 34 = 97

Now, let's solve for S :

S = 97 - 34

S = 63

Step 3: Find B using the value of S from Equation 3

B = S - 26

B = 63 - 26

B = 37