Question

polled biology majors and chemistry majors at the local university to determine how many prefer math to history. her results are shown in the table.

student	prefer math	total students
chemistry majors	16	32

which is a true statement?

• the relationship is proportional because ( 16 - 14 = 2 ) and ( 32 - 30 = 2 )
• the relationship is proportional because ( 30 - 14 = 16 ) and ( 32 - 16 = 16 )
• the relationship is not proportional because ( \frac{14}{32} = \frac{7}{16} ) and ( \frac{16}{30} = \frac{8}{15} )
• the relationship is not proportional because ( \frac{14}{30} = \frac{7}{15} ) and ( \frac{16}{32} = \frac{1}{2} )

Explanation:

Step1: Recall Proportional Relationship

A proportional relationship between two quantities \( y \) (prefer math) and \( x \) (total students) means \( \frac{y}{x} \) is constant (i.e., the ratio of prefer math to total students should be the same for both groups).

Step2: Calculate Ratios for Biology Majors

For biology majors, \( \frac{\text{Prefer Math}}{\text{Total Students}} = \frac{14}{30} \). Simplify: \( \frac{14\div2}{30\div2} = \frac{7}{15} \).

Step3: Calculate Ratios for Chemistry Majors

For chemistry majors, \( \frac{\text{Prefer Math}}{\text{Total Students}} = \frac{16}{32} \). Simplify: \( \frac{16\div16}{32\div16} = \frac{1}{2} \) (or \( \frac{8}{16} \) as in one of the options). Wait, check the options: one option has \( \frac{16}{32}=\frac{8}{16} \)? No, \( \frac{16}{32}=\frac{1}{2} \), but let's check the given options. Wait, the fourth option: \( \frac{14}{30}=\frac{7}{15} \) (correct simplification) and \( \frac{16}{32}=\frac{1}{2} \) (wait, no, \( \frac{16}{32}=\frac{8}{16} \)? Wait, no, \( \frac{16}{32}=\frac{1}{2} \), but let's check the options again. Wait, the third option: \( \frac{14}{32}=\frac{7}{16} \) (wrong, should be \( \frac{14}{30} \)), no. Wait, the fourth option: \( \frac{14}{30}=\frac{7}{15} \) (correct) and \( \frac{16}{32}=\frac{1}{2} \)? Wait, no, \( \frac{16}{32}=\frac{8}{16} \)? No, \( \frac{16}{32}=\frac{1}{2} \), but let's check the logic. Wait, the key is: for proportionality, \( \frac{14}{30} \) should equal \( \frac{16}{32} \). \( \frac{14}{30}=\frac{7}{15}\approx0.4667 \), \( \frac{16}{32}=\frac{1}{2}=0.5 \). Wait, no, wait the fourth option: "The relationship is not proportional because \( \frac{14}{30}=\frac{7}{15} \) and \( \frac{16}{32}=\frac{1}{2} \)"? Wait, no, let's re-express:

Wait, \( \frac{14}{30} = \frac{7}{15} \approx 0.4667 \), \( \frac{16}{32} = \frac{1}{2} = 0.5 \). Wait, but the third option says \( \frac{14}{32}=\frac{7}{16} \) (wrong, numerator is 14, denominator 32? No, biology majors: prefer math 14, total 30. So \( \frac{14}{30} \), not \( \frac{14}{32} \). Wait, maybe a typo in the option. Wait, the fourth option: "The relationship is not proportional because \( \frac{14}{30}=\frac{7}{15} \) and \( \frac{16}{32}=\frac{1}{2} \)"? Wait, no, \( \frac{16}{32}=\frac{8}{16} \)? No, \( \frac{16}{32}=\frac{1}{2} \). Wait, let's check the options again:

Option 1: Difference, but proportionality is about ratio, not difference. Eliminate.

Option 2: Difference, not ratio. Eliminate.

Option 3: \( \frac{14}{32} \) (wrong, should be \( \frac{14}{30} \)) and \( \frac{16}{30} \) (wrong, should be \( \frac{16}{32} \)). Eliminate.

Option 4: \( \frac{14}{30}=\frac{7}{15} \) (correct simplification: 14÷2=7, 30÷2=15) and \( \frac{16}{32}=\frac{1}{2} \) (wait, 16÷16=1, 32÷16=2; or 16÷8=2, 32÷8=4? No, 16/32=1/2. Wait, but the option says \( \frac{16}{32}=\frac{1}{2} \)? Wait, no, the option says \( \frac{16}{32}=\frac{1}{2} \)? Wait, no, the fourth option: "The relationship is not proportional because \( \frac{14}{30}=\frac{7}{15} \) and \( \frac{16}{32}=\frac{1}{2} \)". Wait, \( \frac{7}{15} \approx 0.4667 \) and \( \frac{1}{2} = 0.5 \), which are not equal, so the ratios are different, hence not proportional. Wait, but let's check the option again. Wait, maybe a typo in the option: maybe \( \frac{16}{32}=\frac{8}{16} \)? No, \( \frac{16}{32}=\frac{1}{2} \). Wait, no, the fourth option: "The relationship is not proportional because \( \frac{14}{30}=\frac{7}{15} \) and \( \frac{16}{32}=\frac{1}{2} \)". Since \( \frac{7}{15} eq \frac{1}{2} \), the ratios are not equal, so the relationship is not proportional. So the fourth option is correct (despite possible typo in the option's fraction, but the key is the ratios are different).

Answer:

The relationship is not proportional because \(\frac{14}{30} = \frac{7}{15}\) and \(\frac{16}{32} = \frac{1}{2}\) (the fourth option, assuming the option's fractions are as intended, but based on calculation, the correct reasoning is about unequal ratios). So the correct option is the fourth one: "The relationship is not proportional because \(\frac{14}{30} = \frac{7}{15}\) and \(\frac{16}{32} = \frac{1}{2}\)" (matching the fourth option's description).