Set 6

Explanation

Given the start time is $07:00$ and the observation ends at $09:24$. Let's calculate the total duration in minutes:

It is stated that the three runners passed the start line together $4$ times during this interval. This means they meet every:

This meeting interval corresponds to the Least Common Multiple (LCM) of the runners' lap times.

We need to find the value of $x$ such that $\text{LCM}(12, x, 9) = 36$. First, find the LCM of $12$ and $9$:

Since $\text{LCM}(12, 9)$ is already $36$, for $\text{LCM}(12, 9, x)$ to remain $36$, $x$ must be a factor of $36$ or a number that, when combined with $12$ and $9$, results in an LCM of $36$. Let's check the options:

• If $x = 5$: $\text{LCM}(36, 5) = 180$ (Incorrect)
• If $x = 6$: $\text{LCM}(36, 6) = 36$ (Correct)
• If $x = 7$: $\text{LCM}(36, 7) = 252$ (Incorrect)
• If $x = 10$: $\text{LCM}(36, 10) = 180$ (Incorrect)
• If $x = 13$: $\text{LCM}(36, 13) = 468$ (Incorrect)

Thus, the satisfying value for $x$ is $6$.

Explanation

It is known that runner $A$ takes $12$ minutes to complete $1$ full lap. First, we convert this time into seconds:

The average speed of runner $A$ is $2$ meters per second. We know that speed ($v$) is the ratio of distance ($s$) per unit of time ($t$), as per the formula:

The question asks for the length of the track ($s$):

Converting to kilometers:

Thus, the length of the track covered is $1440$ meters or $1.44$ Km.

Explanation

Given:

• Initial capital ($M_0$) = $\text{Rp}12,500,000.00$
• Final capital ($M_t$) = $\text{Rp}15,000,000.00$
• Saving period ($t$) = $1$ year

First, we calculate the interest amount earned:

The interest percentage is the ratio of the interest amount to the initial capital multiplied by the saving period:

Thus, the money grew due to an interest of $20\%$.

Explanation

Given:

• Initial capital ($M_0$) = $\text{Rp}15,000,000.00$
• Final capital ($M_t$) = $\text{Rp}16,500,000.00$
• Interest rate ($i$) = $12\%$ per year

First, we calculate the value increase (interest earned):

The interest rate is $12\%$ per year. Thus, the duration Bima saved ($t$) can be calculated as:

Converting to months:

Thus, the duration Bima saved is $10$ months.

Explanation

Given:

• Cone A: Radius ($R_1$) is equal to height ($h_1$), so $R_1 = h_1$.
• Cone B: Radius ($R_2$) is twice the height ($h_2$), so $R_2 = 2h_2$ or $h_2 = \frac{1}{2}R_2$.
• Equal volumetric rate means $V_A = V_B$ (Volume of cone A equals Volume of cone B).

Recall the cone volume formula $V = \frac{1}{3}\pi R^2 h$.

Simplify by cancelling $\frac{1}{3}\pi$ on both sides:

Substitute the height-radius relationship for each cone:

• Replace $h_1$ with $R_1$
• Replace $h_2$ with $\frac{1}{2}R_2$

We want to find the ratio $R_1 : R_2$:

Take the cube root of both sides:

Thus, the ratio $R_1 : R_2 = 1 : \sqrt[3]{2}$.

Explanation

Volume Rate Analysis

It is given that the pouring rate is the same for both cones. This means that in the same time $t$, the volume of sand poured is equal.

The formula for the volume of a cone is:

Substituting Radius and Height Relationships

For cone A, given $R_1 = h_1$. Its volume is:

For cone B, given $R_2 = 2h_2$. Its volume is:

Determining the Height Ratio

Since $V_A = V_B$, we have:

Simplify by dividing both sides by $\frac{1}{3}\pi$:

Arrange into a ratio:

Take the cube root of both sides:

Thus, the ratio of heights is $h_1 : h_2 = \sqrt[3]{4} : 1$.

Explanation

Determining the Common Difference

January is the first month ($U_1$), so the number of cars produced is $a = 100$ units. April is the 4th month ($U_4$), with a production of $160$ units.

We can find the common difference ($b$) of this arithmetic sequence using the $n$-th term formula:

Substitute the known values for April ($n=4$):

So, production increases by $20$ units each month.

Calculating Total Annual Production

Total production until the end of the year means the sum of production for $12$ months ($S_{12}$). We use the arithmetic series sum formula:

Substitute $n = 12$, $a = 100$, and $b = 20$:

Thus, the total number of cars that can be produced by the end of the year is $2,520$ units.

Explanation

Determining the Ratio

It is given that the wood is cut into 6 pieces, so $n = 6$. The lengths of the pieces form a geometric sequence.

We assume the first term is the longest (it works the same way if assumed otherwise for the total sum). Let the first term be $a = 96$ and the 6th term be the shortest $U_6 = 3$.

The formula for the $n$-th term of a geometric sequence is:

Substitute the known values:

Simplify the equation to find the ratio ($r$):

So, the ratio between the wood pieces is $\frac{1}{2}$.

Calculating Total Wood Length

The total length of the original wood is the sum of all pieces ($S_6$). Since $r < 1$, we use the formula:

Substitute $a = 96$, $r = \frac{1}{2}$, and $n = 6$:

Calculate the result:

Thus, the total length of the original wooden plank is $189$ cm.

Explanation

Constructing the Mathematical Model

The basic formula for the phone tariff ($y$) is given as:

From the problem, we have the following information:

1. Call duration = $x$ minutes.
2. Subscription fee ($z$) = $60,000$.
3. The rate per minute is $150$ more than the call duration. This means:

Substitution into the Formula

We substitute these components into the initial formula:

Expand the multiplication:

Thus, the function for the landline tariff paid by the customer is $y = x^2 + 150x + 60,000$.

Explanation

Identifying the Concept

This problem involves arranging $3$ people from $7$ candidates to fill specific positions (chairman, secretary, treasurer). Since the order or position matters (chairman $\neq$ secretary $\neq$ treasurer), we use the concept of Permutation.

Calculating Permutation

The formula for the permutation of $r$ objects from $n$ objects is:

Given:

• Number of candidates ($n$) = $7$
• Number of positions chosen ($r$) = $3$

Substitute into the formula:

Expand the factorials:

Thus, the number of possible staff committee arrangements is $210$ ways.

Explanation

Identifying the Concept

This problem asks us to select team members based on their expertise. Since the order of selection does not matter (who is chosen first or second does not change the team composition), we use the concept of Combination.

The combination formula is:

Calculating Mathematician Combinations

From $3$ mathematicians, $2$ will be chosen:

Calculating Physicist Combinations

From $5$ physicists, $3$ will be chosen:

Calculating Total Ways

Since the selection of mathematicians and physicists are independent events occurring together, we multiply the two results:

Thus, there are $30$ ways to form the team.

Explanation

Identifying Proportions

Given:

• To charge from $20\%$ to $100\%$ takes $2.5$ hours.
• The percentage charged is $100\% - 20\% = 80\%$.

Now the battery condition is $60\%$.

• The target charge is $100\%$.
• The percentage needed to charge is $100\% - 60\% = 40\%$.

Direct Proportion

Since the charging power is constant, the time required is directly proportional to the percentage charged.

Substitute the values:

Time Conversion

Convert $1.25$ hours into hours and minutes:

Thus, the time required is 1 hour 15 minutes.

Explanation

Calculating Cost Components

The total cost Bedu has to pay is the sum of the base fare (for the first kilometer) and the total fare for the subsequent kilometers.

Given:

• Total distance = $18$ km.
• Base fare (first 1 km) = $\text{Rp}2,500$.
• Additional fare per km = $\text{Rp}1,000$.

Remaining distance subject to additional fare:

Calculating Total Fare

We calculate the total cost:

Thus, the total fare Bedu has to pay is $\text{Rp}19,500.00$.

Explanation

Defining Variables

Let:

• Salary of Group $A$ be $x$.
• Salary of Group $B$ be $y$.
• Salary of Group $C$ be $z$.

Forming Equations

From the problem, we know:

1. Salary of Group $A$ is one-third of Group $B$'s salary.

   $$x = \frac{1}{3}y \quad \dots (1)$$

2. Salary of Group $C$ is half of Group $B$'s salary.

   $$z = \frac{1}{2}y \quad \dots (2)$$

Solving the Problem

Asked: Income of Group $A$ for $6$ months is equal to income of Group $C$ for $n$ months.

The equation is:

Substitute equations $(1)$ and $(2)$ into the equation above (express everything in terms of $y$):

Since $y \neq 0$, we can divide both sides by $y$:

Thus, the income will be equal to the income of Group $C$ employees for $4$ months.

Explanation

Defining Variables

Let:

• Number of type $\text{I}$ trucks be $x$.
• Number of type $\text{II}$ trucks be $y$.
• Number of type $\text{III}$ trucks be $z$.

Forming System of Equations

From the problem, we know:

1. The total number of trucks rented is $20$.

   $$x + y + z = 20 \quad \dots (1)$$

2. The total carrying capacity is $96$ kg, with capacities for each type being $3$, $4$, and $6$ kg, respectively.

   $$3x + 4y + 6z = 96 \quad \dots (2)$$

3. It is known that the company rents $4$ type $\text{I}$ trucks.

   $$x = 4$$

Solving the System

Substitute $x = 4$ into equation $(1)$:

Substitute $x = 4$ into equation $(2)$:

Simplify the equation above by dividing both sides by $2$:

Eliminate variable $z$ from equations $(3)$ and $(4)$. Multiply equation $(3)$ by $3$:

Subtract the second equation from the first:

Thus, the number of type $\text{II}$ trucks rented is $6$.

Explanation

Calculating Initial Total Weight

The initial average weight $(\bar{x}_1)$ of ten players $(n_1 = 10)$ is $60$ kg. The initial total weight can be calculated as follows:

Calculating the Weight of the Replaced Player

Let the weight of the replaced player be $n$ and Iksan's weight be $x_i = 62$ kg.

After the replacement, the number of players remains $10$, but the average weight becomes $\bar{x}_2 = 60.5$ kg. The equation is:

Substitute the known values:

Multiply both sides by $10$: