Q: A group of mountaineers knows that for every 100 m increase in altitude, the temperature drops by 1°C. If they are at 1,000 m with a temperature of 20°C, which expression allows them to calculate the temperature at 4,000 m?

$$\text{Temperature} = -\frac{\text{Altitude}}{100} + 30$$

Question 1

A company offers a basic computer training program consisting of 4 modules. Module I has an intensity of 40 hours and costs $$35,000 per hour. If the company pays 4$$,200,000 to train all employees of the 'Insumos' department in Module I, how many employees are in this department?

Accepted Answer

Between 61 and 90 employees.

Question 2

A person invests in dollars in the United States, knowing that the exchange rate is constant at 1 USD = 4,000 COP, and their investment will yield a 3% gain in dollars this month. A friend claims that the gain in Colombian pesos will also be 3%. Is this statement correct?

Accepted Answer

Yes, because 3% represents a fixed proportion in both currencies, since the exchange rate remains constant.

Question 3

A school wants to organize a museum trip for 140 students, dividing them into 3 groups. The available time slots and their maximum capacities and prices are: Slot 1: 50 students at $$35,000; Slot 2: 40 students at 40$$,000; Slot 3: 30 students at $$50,000; Slot 4: 60 students at 45$$,000. If the school chooses slots 1, 3, and 4, does this guarantee all students can attend at the lowest possible price?

Accepted Answer

No, because a lower price is possible by choosing slot 2 instead of slot 3.

Question 4

A group of mountaineers knows that for every 100 m increase in altitude, the temperature drops by 1°C. If they are at 1,000 m with a temperature of 20°C, which expression allows them to calculate the temperature at 4,000 m?

Accepted Answer

$$	ext{Temperature} = -\frac{	ext{Altitude}}{100} + 30$$

Question 5

A rectangular field has its length doubled (increased by 100%) while the width remains the same. What happens to its area?

Accepted Answer

It doubles.

Question 6

A bookstore offers a 10% discount on all printed books. Juan has an additional 10% discount coupon, applied to the already discounted price. To find the final price of a book originally costing $50,000, Juan does the following: Step 1: Multiply the original price by 9. Step 2: Divide the result by 10. Step 3: Multiply the result by 10. Step 4: Divide the result by 100. In which step is there an error?

Accepted Answer

Step 3, because the final price is 90% of the sale price, so he should multiply by 90, not 10.

Question 7

A hotel assigns 4-digit extension numbers to rooms using only the digits 2, 3, 4, and 5, with no repeated digits. How many different extensions are possible?

Accepted Answer

24

Question 8

A company sells tables for $$40,000 and chairs for 20$$,000. The owner knows that total sales were $$1,400,000 and that three times as many chairs as tables were sold. Let M$ be the number of tables and S$ the number of chairs sold. Which system of equations correctly models the situation?

Accepted Answer

No, because although equation 1 relates each price to the correct variable, equation 2 means three times as many tables as chairs were sold.

Question 9

A pizza restaurant had 110 customers on Sunday. 20 chose a hot drink, 40 chose a cold drink, 100 chose pizza, and no one chose both hot and cold drinks. Which of the following can be calculated with this information?

Accepted Answer

The total number of customers who chose only pizza.

Question 10

A sound system's intensity level is increased by 20 dB. According to the rule that a 10 dB increase means multiplying the intensity by 10, by what factor is the intensity multiplied for a 20 dB increase?

Accepted Answer

By 100.

Question 11

A password system requires users to enter a 3-character password using the 26 letters (upper and lower case) and 10 digits (0-9). Consider these restrictions:
R1: Any three digits.
R2: Three different digits.
R3: One uppercase letter, one lowercase letter, one digit (in that order).
R4: Any letter (upper or lower case), then two digits.
Order the restrictions from fewest to most possible passwords.

Accepted Answer

R2, R1, R4, R3.

Question 12

A triangle has sides of 7 m, 7 m, and 18 m. Is it possible for this triangle to have a right angle and an angle less than 45°?

Accepted Answer

No, because the triangle cannot have sides 7, 7, and 18 m.

Question 13

A real estate developer sells apartments at $$800,000 per square meter, with an annual appreciation of 5%. Which expression gives the price per square meter after n$ years?

Accepted Answer

$$800,000 \left( 1 + \frac{5}{100} \right)^n$$

Question 14

A cylindrical container has height $$h = 2$$ and base radius $$r = \frac{3}{2}. A $$box has a square base with side $$L = \frac{3}{2}$$ and height $$h = 2. A $$sphere has radius $$r = \frac{3}{2}. $$Which statement is true about their volumes?

Accepted Answer

The cylinder's volume is greater than the box's; the sphere's volume equals the cylinder's.

Question 15

A person claims that when drawing a ball at random from a bag containing 3 red, 3 black, and 12 white balls, each color has the same probability of being chosen. Is this statement true?

Accepted Answer

No, because there are more balls of one color than the others.

Question 16

A triangle has a base and a height $$h = 120 cm. $$The height is calculated using $$\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.71. If $$the height is cut in half, at approximately what distance from the base is the cut made?

Accepted Answer

42 cm

Question 17

A sound has an intensity $$I$$ and the minimum audible intensity is $$I_0. If$$ $$10^3$$ = \frac{I}{$$I_0$$}$$, how many times greater is I$ than I_0$?

Accepted Answer

A thousand times.

Question 18

A group of 5 students is to be sent to a forum: 2 from grade 10 (out of 5 candidates) and 3 from grade 11 (out of 4 candidates). How many different groups can be formed?

Accepted Answer

40

Question 19

A person wants to construct rational numbers greater than an integer $$k$$ and as close as possible to $$k$$, using $$k + \frac{1}{j}$$ for positive integers $$j. $$How many rational numbers less than $$k + \frac{1}{11}$$ can be constructed this way?

Accepted Answer

An infinite number, since there are infinitely many integers greater than 11.

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