Let’s take advantage of today’s Halloween to enjoy some math problems! We’ll start with easier ones and gradually increase the difficulty. Hold on tight, because things are about to get tricky!
The Money Problem
Statement
The Méndez family wants to decorate their house for Halloween with a budget of 450€. Together, they’ve made a list of what they want to buy:
- 2 strings of lights
- 200 rubber spiders
- 1 projector
- 300 candies
- 4 fabric ghosts
- 3 light bulbs
- 5 costumes
They head to the mall and find the following prices:
- A string of lights: 5.99€ (with an 80% discount)
- A bag of 100 rubber spiders: 3.99€
- A projector: 49.99€
- A bag of 25 assorted candies: 9.99€ (with a 70% discount)
- A fabric ghost: 4.99€
- A light bulb: 14.99€
- A costume: 21€ (with a 65% discount)
They write down the prices and return the next day… but now the discounts are gone!
1. Calculate how much everything would cost without the discounts. Will they be able to afford it with their set budget?
After seeing the prices have increased, they decide to purchase everything and head to the counter, where the cashier informs them that a discount is available for the three cheapest items at 50% off (applicable to all units of these three cheapest products), but the most expensive item will cost them double (applicable to all units of the most expensive product).
2. Explain why they should or should not take the discount.
Finally, they decide to drop the most expensive item and no longer buy it:
3. Should they take the discount now?
Solution
Part 1
First, let’s calculate how much everything would cost without discounts. We’ll calculate the price of each product without any discounts:
- A string of lights: 5.99€ · 100 / 80 = 599 / 80 ≈ 7.48€
- A bag of 100 rubber spiders: 3.99€
- A projector: 49.99€
- A bag of 25 assorted candies: 9.99€ · 100 / 70 = 999 / 70 ≈ 14.27€
- A fabric ghost: 4.99€
- A light bulb: 14.99€
- A costume: 21€ · 100 / 65 = 2100 / 65 ≈ 32.31€
Now, the family wants to buy multiple units of most of these products, so let’s multiply:
- 2 strings of lights: 7.48 · 2 = 14.96€
- 200 rubber spiders (2 bags of 100): 3.99€ · 2 = 7.98€
- 1 projector: 49.99€
- 300 candies (12 bags of 25): 14.27€ · 12 = 171.24€
- 4 fabric ghosts: 4.99€ · 4 = 19.96€
- 3 light bulbs: 14.99€ · 3 = 44.97€
- 5 costumes: 5 · 32.31 = 161.55€
Now, let’s sum everything up to see the total cost:
Total = 14.96 + 7.98 + 49.99 + 171.24 + 19.96 + 44.97 + 161.55 = 470.65€
So, they cannot afford it within their budget of 450€.
Solution: They will have to pay 470.65€, so they can’t afford it with their budget.
Part 2
Now, let’s look at Part B, which will be faster now that we’ve solved Part A. First, we need to identify the three cheapest products. These are: a string of lights, a bag of rubber spiders, and a fabric ghost. We will apply the 50% discount to these three items:
- 2 strings of lights: 14.96€ · 0.5 = 7.48€
- 200 rubber spiders: 7.98€ · 0.5 = 3.99€
- 4 fabric ghosts: 19.96€ · 0.5 = 9.98€
Next, we find the most expensive product, which is the projector. We’ll apply a double price to it:
- 1 projector: 49.99€ · 2 = 99.98€
Now, let’s sum the new prices of these products and the prices of the others:
Total = 7.48 + 3.99 + 9.98 + 99.98 + 19.96 + 44.97 + 161.55 = 499.19€
By taking the discount, they will actually pay more than they would without it.
Solution: They should not take the discount, as it will make them pay much more than without it.
Part 3
Now, let’s consider the case where they decide not to buy the most expensive item. To do this, we subtract the cost of the projector at double price (99.98€) from the total in Part B (499.19€):
499.19€ - 99.98€ = 399.21€
Next, we consider the second most expensive product, which is the costume. We will double the cost for all five costumes:
- 5 costumes: 5 · 32.31 = 161.55€ × 2 = 323.10€
Now, we add the new cost of the costumes (323.10€) to the remaining total:
399.21€ + 323.10€ = 722.31€
Clearly, this is much worse than before.
Solution: They should not take the discount, as they would pay significantly more.
The Transportation Problem
Statement
Six friends have gathered to go trick-or-treating. Two of them live on the same street, two live on another street (quite far from the first), and the last two live on another street (very far from the other two). They decide to meet at the town square, so they need to take some form of transportation. The two friends who live together use the same mode of transportation. Based on the following statements, and assuming they are all true, what mode of transportation does Tomás use?
- Alejandro still doesn’t have a driver’s license and goes with Benito, who does not take the bus.
- Andrés takes the bus.
- Carlos does not go with Darío, nor does he take the bus.
- One couple goes by train, another by bus, and the third by car.
Solution
To solve this, let’s first examine the first statement:
- Alejandro doesn’t drive, so he is not going by car.
- Benito is not taking the bus.
- Alejandro and Benito must be the couple going by train.
Next, the second statement gives us that Andrés is on the bus.
At this point, we know three people are assigned transportation: the train (Alejandro and Benito), and the bus (Andrés). Now, based on the third statement, we know that Carlos is not on the bus, and he is not with Darío, so he must be taking the car.
Now, we have two people left: Darío and Tomás. The only mode of transportation left for Darío is the bus, and therefore, Tomás must be the one using the car.
Solution: Tomás is using the car.
The Candy Problem
Statement
The average number of candies collected in a class of 20 students was 6 candies each. 8 students collected 3 candies each. What is the average number of candies collected by the rest of the class?
Solution
First, we calculate the total number of candies collected by all 20 students, knowing that the average number of candies per student is 6:
20 · 6 = 120 candies
Now, we calculate how many candies the 8 students who collected 3 candies each took:
8 · 3 = 24 candies
Next, we subtract the number of candies collected by the 8 students from the total number of candies:
120 – 24 = 96 candies
Finally, we calculate the average number of candies collected by the remaining 12 students:
96 / 12 = 8 candies
Solution: The remaining students collected, on average, 8 candies each.
