Set 7

Explanation

Determining Meeting Interval

It is known that the three vans meet for the $4$-th time $18$ hours after the initial meeting.

Meeting for the $4$-th time means that $3$ meeting intervals (LCM) have passed.

Let $T$ be the joint meeting interval (LCM of periods A, B, and C).

So, the three vans meet together every $6$ hours.

Determining Value of x

The joint meeting interval ($T$) is the Least Common Multiple (LCM) of the periods of each van.

Known periods:

• Van A: $2$ hours
• Van B: $3$ hours
• Van C: $x$ hours

Thus:

We know that $\text{LCM}(2, 3) = 6$. Therefore, we must find the value of $x$ such that $\text{LCM}(6, x) = 6$. For $\text{LCM}(6, x) = 6$, $x$ must be a factor of $6$.

From the available options, we are looking for a value of $x$ that is a factor of $6$.

The values $5$, $7$, $8$, and $9$ are not factors of $6$, so they do not satisfy the condition $\text{LCM}(6, x) = 6$.

For $x = 6$, we get $\text{LCM}(6, 6) = 6$, which satisfies the condition.

So, the satisfying value of $x$ is $6$.

Explanation

Calculating Loan Interest

Given:

• Principal loan ($M$) = $\text{Rp}3,000,000.00$
• Monthly interest ($i$) = $2\%$
• Loan duration ($n$) = $10$ months

The total interest amount is:

Calculating Monthly Installment

The total amount to be paid is the principal plus total interest:

Since the repayment period is $10$ months, the monthly installment is:

So, the money Panjul must deposit per month is $\text{Rp}360,000.00$.

Explanation

Here is the calculation of sales for each item (Recall that $1$ dozen = $12$ pieces/copies).

Notebooks

The purchase capital for $6$ dozens of notebooks is $\text{Rp}144,000.00$.

Sales:

• $3$ dozens ($36$ pieces) sold at $\text{Rp}2,500.00$ each
• $2$ dozens ($24$ pieces) sold at $\text{Rp}2,300.00$ each
• Remaining $1$ dozen ($12$ pieces) sold at $\text{Rp}2,100.00$ each

Total revenue from notebooks:

Profit from notebooks:

Ballpoints

The purchase capital for $5$ dozens of ballpoints is $\text{Rp}90,000.00$.

Sales:

• $1$ dozen ($12$ pieces) sold at $\text{Rp}2,000.00$ each
• $3$ dozens ($36$ pieces) sold at $\text{Rp}1,700.00$ each
• Remaining $1$ dozen ($12$ pieces) sold at $\text{Rp}1,600.00$ each

Total revenue from ballpoints:

Profit from ballpoints:

Pencils

The purchase capital for $6$ dozens of pencils is $\text{Rp}72,000.00$.

Sales:

• $4$ dozens ($48$ pieces) sold at $\text{Rp}1,500.00$ each
• Remaining $2$ dozens ($24$ pieces) sold at $\text{Rp}1,300.00$ each

Total revenue from pencils:

Profit from pencils:

Conclusion

Profit comparison:

• Notebooks: $\text{Rp}26,400.00$
• Ballpoints: $\text{Rp}14,400.00$
• Pencils: $\text{Rp}31,200.00$

So, the item that provides the most profit is pencils.

Explanation

Problem Visualization

Here is an illustration of the spotlight positions (at the top corners) and the intersection point of their beams.

Geometric Analysis

The spotlights are installed at the four top corners of the room (points $E$, $F$, $G$, $H$ in the figure). The beams meet at one point. Since the room is cube-shaped and the light positions are symmetrical, this intersection point must be exactly in the center of the cube (the intersection of the space diagonals).

The distance from the center of the cube to the floor (plane $ABCD$) is half of the cube's height (edge length).

Given the edge length $s = 5$ meters.

Then the distance from the intersection point to the floor is:

So, the distance from the intersection point to the floor is $2\frac{1}{2}$ meters.

Explanation

Volume Analysis

The remaining water volume is the difference between the cylinder volume (initial water) and the total volume of the balls inserted.

Substitute $r_1 = 16 \text{ cm}$ and $t = 20 \text{ cm}$ for the cylinder, and $r_2 = 6 \text{ cm}$ for the ball.

Simplify the equation by dividing both sides by 32:

Since $n$ represents the number of balls, $n$ must be an integer. Furthermore, the total volume of the balls cannot exceed the capacity that allows for $64\pi$ of remaining water (if $n=18$, the remaining water would be negative, which is impossible).

Therefore, the number of balls inserted is $17$ balls.

Explanation

Arithmetic Sequence Analysis

The pattern of temperature decrease in the table forms an arithmetic sequence. For every $100 \text{ m}$ increase in altitude, the temperature drops by $2^\circ\text{C}$.

Given:

• The first term ($U_1$) at an altitude of $100 \text{ m}$ is $32$.
• The common difference ($b$) is $-2$.

An altitude of $800 \text{ m}$ corresponds to the 8th term ($U_8$), and an altitude of $1200 \text{ m}$ corresponds to the 12th term ($U_{12}$).

The formula for the $n$-th term of an arithmetic sequence is:

So, the ratio of the air temperature at the two altitudes is:

Therefore, the temperature ratio is $9:5$.

Explanation

Arithmetic Sequence Identification

Based on the information, Mr. Brikson's salary forms an arithmetic sequence with:

• Salary this year (age $40$) as the first term ($a$): $a = 15.750.000$
• Annual salary increase as the common difference ($b$): $b = 750.000$

We need to find the salary at age $54$. The difference in years from age $40$ to $54$ is:

Since the salary increases every year, the salary at age $54$ is the initial salary plus $14$ increases.

Salary Calculation

Using the formula for the $n$-th term (where $n$ here represents the 15th year of the sequence, i.e., after 14 increases):

Substituting the values:

So, Mr. Brikson's salary when he is $54$ years old is $\text{Rp}26.250.000,00$.

Explanation

Mathematical Modeling

Known:

• Income ($y$) = UMR + (number of passengers $\times$ satisfaction index)
• Number of passengers = $x$
• Satisfaction index ($z$) = $x - 100$
• UMR = $\text{Rp}4.680.000,00$

So the mathematical model is:

Income Calculation

If the number of passengers is $x = 200$, substitute it into the equation:

So, the driver's income is $\text{Rp}4.700.000,00$.

Explanation

Position Equation

The collision event occurs when the position function with respect to time, for both the rocket and the ship, has the same value.

Factor the quadratic equation:

Since time cannot be negative, the positive value is used. This means they collide when $t = 40$ seconds.

Explanation

Identify Information

Given:

• Total inventory items $n(S) = 80$
• Damaged items $= 20$

Thus, the number of undamaged items $n(A)$ is:

Probability Calculation

The probability of picking an undamaged item is:

So, the probability that the item taken is not damaged is $\frac{3}{4}$.

Explanation

First Box Probability

The first box contains $4$ red balls and $3$ blue balls. The total number of balls in the first box is:

The probability of drawing a red ball from the first box ($M$) is:

Second Box Probability

The second box contains $7$ red balls and $3$ white balls. The total number of balls in the second box is:

The probability of drawing a white ball from the second box ($P$) is:

Combined Probability

Since drawing balls from the two boxes are independent events, the probability of drawing a red ball from the first box and a white ball from the second box is:

Thus, the probability is $\frac{12}{70}$.

Explanation

Calculating Actual Distances

The map scale is $1 : 2,000,000$. The actual distance ($J_s$) can be calculated using the formula:

Actual Distance from City $A$ to City $x$:

Actual Distance from City $x$ to City $B$:

Actual Distance from City $A$ to City $y$:

Actual Distance from City $y$ to City $B$:

Calculating Travel Distances

Benu's Journey ($A \to x \to B$):

Mamang's Journey ($A \to y \to B$):

Calculating Distance Difference

The difference in travel distance between Benu and Mamang is:

Thus, the difference in travel distance is $60 \text{ km}$.

Explanation

Track Analysis

The track is circular with $4$ checkpoints: $A, B, C, D$. A full lap consists of $4$ segments: $A \to B$, $B \to C$, $C \to D$, and $D \to A$. The journey starts from $A$.

Calculating Initial Distance

The runner passes checkpoint $C$ $3$ times.

• 1st time: $A \to B \to C$ ($2$ segments).
• 2nd time: Add 1 full lap ($4$ segments). Total: $2 + 4 = 6$ segments.
• 3rd time: Add 1 full lap ($4$ segments). Total: $6 + 4 = 10$ segments.

It is known that the travel time for $10$ segments is $12$ minutes. The time per segment ($t_s$) is:

Calculating Total Time

To reach checkpoint $C$ for the 4th time, the runner must cover 1 more lap ($4$ segments). The total segments covered are:

Thus, the total time required is:

Therefore, the time required by the runner is $16.8 \text{ minutes}$.

Explanation

Creating Mathematical Model

Let:

• $A$ = capital amount for product $A$
• $B$ = capital amount for product $B$

Given the total capital is $\text{Rp}10,000,000.00$:

Given the total profit is $\text{Rp}904,000.00$ with product $A$ profit of $8\%$ and product $B$ profit of $10\%$:

Multiply both sides by $10$ to simplify the calculation:

Solving the System of Equations

Eliminate variable $B$ from equations $(1)$ and $(2)$:

Calculate the value of $A$:

Therefore, the capital for product $A$ is $\text{Rp}4,800,000.00$.

Explanation

Creating a Mathematical Model

Let:

• $x$ = price of meat per kg
• $y$ = price of fish per kg

Based on the table, we can create a system of two linear equations.

For Store A:

For Store B:

Finding the Price of Fish

We will find the value of $y$ (price of fish). Eliminate variable $x$ from equations $(1)$ and $(2)$.

Multiply equation $(1)$ by $7$ and equation $(2)$ by $4$ so that the coefficients of $x$ are the same:

Subtract the two equations:

Therefore:

Since the unit in the table is "thousand rupiah", the price of fish is $20$ thousand rupiah or $\text{Rp}20,000.00$.

Thus, the price of fish per kilogram is $\text{Rp}20,000.00$.

Explanation

Initial Data Analysis

Given:

• Initial number of participants: $n_1 = 22$
• Initial average: $\bar{x}_1 = 5$
• Range: $R = 4$

Let the scores be sorted from smallest to largest as $x_1, x_2, \dots, x_{22}$.

The sum of all initial scores is:

The range is defined as the difference between the largest and smallest values: