Everyday, Dave eats either a sandwich or pizza for lunch. Over 42 days, Dave had pizza 3 times for every 4 times he had a sandwich. Over the next x days, he had pizza 3 times for every 2 times he had a sandwich. If at the end of this entire period he had pizza as many times as he has a sandwich, what is the value of x? Please explain.

He must eat either 1 pizza a day or 1 sandwich a day. He can't eat 2 of anything during 1 day. So he can't eat 2 pizzas in a day or 2 sandwiches in a day.

Step One

Find the number of days in a cycle.

3 out of 4 days means out of 4 days he chooses Pizza

2 out of 3 days means out of 3 days he chooses a Sandwich.

That means that there are 4 + 3 days in total = 7 days in the cycle.

Step Two

Find the number of Pizzas and the number of Sandwiches he had during the 42 day period.

3/7 * 42 = Pizzas

3*6 = 18 pizzas in 42 days.

4/7* 42 = Sandwiches

4*6 = 24 sandwiches.

In 42 days he ate 24 + 18 = 42 lunches of either pizzas or sandwiches.

Step three This is the key step.

Find the ratio of sandwiches and pizzas for x days.

Every 5 days he either has 3 pizzas or 2 sandwiches. If the total number of days is x then

3/5 * x = Pizzas

2/5 * x = Sandwiches.

Step 4

Equate the results.

Pizzas = sandwiches.

18 + 3/5 x = 24 + 2/5 x Multiply by 5

18*5 + (3/5)*x * 5 = 24 * 5 + (2/5)*x*5

90 + 3x = 120 + 2x Subtract 2x from both sides

90 + 3x - 2x = 120

90 + x = 120 Subtract 90 from both sides.

x = 120 - 90

x = 30 days.

Conclusion

It takes 30 days to even up the number of pizzas with the number of sandwiches.