Using your monthly and yearly counts, apply the 30% and 70% rates to find how many monthly members and how many yearly members are enrolled in automatic payment.

Use the counts you found to check: Does the total number of automatic-payment members double in the hypothetical scenario? Is the total more than half? What happens to the overall proportion if only the monthly rate increases?

When a statement talks about one group being "more likely" than another, it is asking you to compare the percentages of those groups who enroll in automatic payment, not just the raw numbers.

In Desmos, type 0.8*0.3 + 0.2*0.7 and look at the numeric result. This is the overall fraction of new members who are in automatic payment under the original situation.

To model the situation where all members chose yearly memberships, type 0.7 (representing 70% automatic payment for all). Compare this to the value from step 1 and decide whether the new value is exactly double the original.

To model increasing the monthly automatic-payment rate from 30% to 40% while keeping the yearly rate at 70%, type 0.8*0.4 + 0.2*0.7. Compare this new overall proportion with the original value from step 1 to see whether the overall proportion stays the same or changes.

To make the percentages concrete, imagine there are 100 new members in total.

• 80% chose monthly: that is $80$ members.
• 20% chose yearly: that is $20$ members.

We will now use these counts to figure out how many enrolled in automatic payment and then test each answer choice.

Use the given percentages within each group:

• Monthly: 30% of the 80 monthly members enrolled in automatic payment.
  • $0.30\times 80 = 24$ monthly members in automatic payment.
• Yearly: 70% of the 20 yearly members enrolled in automatic payment.
  • $0.70\times 20 = 14$ yearly members in automatic payment.

So out of 100 new members, $24+14=38$ are enrolled in automatic payment.

Now use the numbers to test some of the options.

• Current total in automatic payment: 38 out of 100, or 38%.

• Choice C ("More than half..."): More than half would mean more than 50 out of 100. We only have 38, so C is not supported.

• Choice A (everyone becomes yearly): If all 100 members were on yearly plans at a 70% automatic-payment rate, then $0.70\times 100 = 70$ members would be in automatic payment.

  • Doubling the current 38 would be $2\times 38 = 76$.
  • 70 is less than 76, so the total would not double. So A is not supported either.

Choice D says: if the monthly automatic-payment percentage increased by 10 percentage points (from 30% to 40%), then the yearly automatic-payment percentage would also need to increase to keep the overall proportion the same.

Compute the new overall proportion if monthly went from 30% to 40% while yearly stays at 70%:

• Monthly: $0.40\times 80 = 32$ members.
• Yearly: still $0.70\times 20 = 14$ members.
• New total in automatic payment: $32+14=46$ out of 100, or 46%.

Originally, the overall proportion was 38%; after raising the monthly rate, it became 46% even though the yearly percentage did not increase. That means you do not need to raise the yearly percentage to keep the same overall proportion (in fact, keeping it the same did not keep the proportion unchanged). So D is not supported.

We have all the information we need to evaluate the remaining type of statement: how likely each group is to enroll in automatic payment.

• Among monthly members, 30% enroll in automatic payment.
• Among yearly members, 70% enroll in automatic payment.

Because $70\% > 30\%$, a randomly chosen yearly member is more likely to be in automatic payment than a randomly chosen monthly member.

This matches statement B: Members who chose the yearly membership were more likely to enroll in automatic payment than members who chose the monthly membership.