Matthew would like to join an online book club that costs $84 per year. If he signs up for an automatic renewal, he will get 5% off his membership. If he signs up in July, he can get an additional 20% off. In addition, for every 2 people he refers to the club, he can get $5 off his membership. How many people does Matthew refer if he pays less than $48 for his membership? Explain.

5% of $84 = $4.20

20% of $84 = $16.80

Assuming no discounts, he needs n referrals, so that

84 - 5n < 48

Adjust that 84 according to how many discounts he achieved.

To determine how many people Matthew refers in order to pay less than $48 for his membership, we need to consider all the discounts and find the total cost after each discount is applied.

First, we'll determine the cost after the automatic renewal discount of 5%. The cost of the membership without any discount is $84 per year. So, the discount amount would be 5% of $84, which is $4.20. Therefore, after the automatic renewal discount, the cost will be $84 - $4.20 = $79.80 per year.

Next, we'll calculate the cost after the additional 20% discount for signing up in July. The discount amount would be 20% of $79.80, which is $15.96. So, the cost after the July discount will be $79.80 - $15.96 = $63.84 per year.

Now, we need to determine how many referrals Matthew needs to get his membership cost below $48. Let's assume he refers 'x' number of people. For each referral, he gets $5 off his membership. So, the total discount he receives from referrals will be 5x.

To find the final cost after the referral discount, we subtract the total referral discount from the cost after the July discount: $63.84 - 5x.

Since we want the cost to be less than $48, we can write an inequality:
$63.84 - 5x < $48.

Now we can solve this inequality to find the values of x:
63.84 - 5x < 48
-5x < 48 - 63.84
-5x < -15.84
Divide both sides by -5, remember to flip the inequality sign when dividing by a negative number since we are dividing by -5:
x > -15.84/-5
x > 3.168

Since we cannot have a fraction of a referral, the minimum number of people Matthew needs to refer is 4.