Set 5

Explanation

Let the price of one belt be $x$ and the price of one tie be $y$. Based on the information provided, we can form the following system of linear equations:

To find the values of $x$ and $y$, we can use the elimination method. We will eliminate the variable $x$ by multiplying the second equation by $3$ so that the coefficient of $x$ becomes the same in both equations.

Next, we subtract the first equation from the modified second equation to solve for $y$.

Having found $y = 12,500$, we substitute this value into one of the original equations, for example $x + 3y = 62,500$, to find $x$.

Thus, the price of one belt is $\text{Rp }25,000.00$ and the price of one tie is $\text{Rp }12,500.00$. Since Yunus wants to buy $1$ of each, the total amount he needs to pay is the sum of both prices.

Therefore, Yunus has to pay $\text{Rp }37,500.00$.

Explanation

To determine when all three suppliers deliver vegetables simultaneously, we use the concept of the Least Common Multiple (LCM) of their delivery schedules, which are $x$, $4$, and $7$.

It is given that the $16$-th simultaneous delivery occurs $1260$ days after the first delivery. This means that $15$ simultaneous delivery intervals have passed since the first meeting.

We can calculate the LCM of the three numbers by dividing the total number of days by the number of intervals occurred:

So, the LCM of $x$, $4$, and $7$ is $84$. We know that the LCM of $4$ and $7$ is $28$. Therefore, we must find the value of $x$ such that the LCM of $x$ and $28$ is $84$.

Let's check the available options:

• If $x = 6$, then LCM($6, 28$) = $84$. (Correct)
• If $x = 8$, then LCM($8, 28$) = $56$. (Incorrect)
• If $x = 9$, then LCM($9, 28$) = $252$. (Incorrect)
• If $x = 10$, then LCM($10, 28$) = $140$. (Incorrect)
• If $x = 11$, then LCM($11, 28$) = $308$. (Incorrect)

Thus, the value of $x$ that satisfies the condition is $6$.

Explanation

April $2024$ has $30$ days. Deliveries start on day $1$, so supplier B delivers on days $1, 5, 9, 13, 17, 21, 25, 29$ for a total of $8$ deliveries. Supplier C delivers on days $1, 8, 15, 22, 29$ for a total of $5$ deliveries.

Thus, the total weight from suppliers B and C during April $2024$ is $246 \text{ kg}$.

Explanation

The television price before tax is $\text{Rp }4,500,000.00$. A $10\%$ tax is applied to the base price, so the tax amount is $\text{Rp }450,000.00$. After tax, the price becomes $\text{Rp }4,950,000.00$.

The $5\%$ discount is applied to the taxed price, so the discount amount is $\text{Rp }247,500.00$. The amount that must be paid is:

Thus, the amount Mrs. Juju must pay is $\text{Rp }4,702,500.00$.

Explanation

This problem can be solved using the concept of a geometric sequence, where bacteria divide into two every $24$ hours (1 day).

For the first 5 days (day 1 to day 5), the number of bacteria follows a geometric sequence with an initial term $a = 10$ and a ratio $r = 2$. The number of bacteria on day 5 is:

Every $120$ hours (5 days), half of the total bacteria are killed. Thus, the remaining bacteria at the end of day 5 is:

For the second 5 days (day 6 to day 10), the cycle restarts with an initial count of $80$. On day 6, the number of bacteria becomes $80 \times 2 = 160$. Thus, the number of bacteria on day 10 (the 5th term of the second phase) is:

At the end of day 10, half of the bacteria are killed again:

Next, we calculate the growth for day 11 and day 12:

Thus, the number of bacteria on the 12th day of observation is $5120$.

Explanation

It is given that the ages of Pak Asep's children form an arithmetic progression with the number of children $n = 6$. Let $U_n$ be the age of the $n$-th child.

From the problem information:

• The age of the $3$rd child is $18$ years: $U_3 = 18$.
• The age of the $5$th child is $16$ years: $U_5 = 16$.

In an arithmetic progression, the formula for the $n$-th term is $U_n = a + (n-1)b$. We can form the equations:

Subtract the first equation from the second to find the common difference ($b$):

Substitute $b = -1$ into the equation for $U_3$ to find the first term ($a$):

So, the age of the first child is $20$ years, and the age difference is $-1$ year. The ages of the six children are: $20, 19, 18, 17, 16, 15$.

The sum of all children's ages ($S_6$) can be calculated using the arithmetic series formula:

Thus, the sum of the ages of all Pak Asep's children is $105$ years.

Explanation

We use the concept of the mean (average) value ($\bar{x}$), which is defined as the sum of all data values ($\Sigma x$) divided by the number of data points ($n$).

Initial Condition

Initially, there are $39$ students with an average score of $45$. Their total score is:

After Yanto Joins

When Yanto joins, the number of students increases to $n_2 = 39 + 1 = 40$. The new class average becomes $46$. The new total score is:

Finding Yanto's Score

Yanto's score ($Y$) is the difference between the new total score and the initial total score.

Thus, Yanto's exam score is $85$.

Explanation

To find the combined average of all classes, we need to calculate the total score of all students and divide it by the total number of students.

Step 1: Calculating Total Scores for Each Class

Class IPA ($n_1 = 30$, $\bar{x}_1 = 6.5$):

Class IPS ($n_2 = 25$, $\bar{x}_2 = 7$):

Class Bahasa ($n_3 = 20$, $\bar{x}_3 = 8$):

Step 2: Calculating Combined Average

The total score of all classes is:

The total number of students in all classes is:

Thus, the combined average ($\bar{x}_{total}$) is:

So, the average score of all these class 12 students is $7.07$.

Explanation

We will use the multiplication rule (filling slots) to form the two-digit house numbers.

Identifying Available Digits

Whole numbers less than $10$ are $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. The condition is that all digits must be odd. Therefore, the allowed digits are:

There are $5$ odd digits available. Since the house number consists of two digits, we have two positions to fill:

1. First Digit: Can be chosen from the $5$ odd digits.
2. Second Digit: Can also be chosen from the $5$ odd digits (repetition is allowed).

Calculating Total Possibilities

The total number of possibilities is the product of the choices for each position:

So, there are $25$ possible house numbers.

Explanation

To determine the number of possible house numbers, we can use the multiplication rule (filling slots).

Step 1: Identifying Available Digits

The available digits are whole numbers less than $10$, which are $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

From this set, we categorize them into:

• Odd Digits: $\{1, 3, 5, 7, 9\}$ (there are $5$ digits).
• Even Digits: $\{0, 2, 4, 6, 8\}$ (there are $5$ digits).

Step 2: Filling the Digit Positions

The house number consists of two digits with the following conditions:

1. First Digit (Odd): We must choose one of the $5$ available odd digits.
2. Second Digit (Even): We must choose one of the $5$ available even digits.

Step 3: Calculating Total Possibilities

The total number of ways to form the house number is the product of the choices for each position:

So, there are $25$ possible house numbers that can be formed.

Explanation

The core concept here is maintaining the consistency of the tea's taste. For the tea to taste the same, the ratio between tea bags and sugar must remain constant, regardless of how many glasses are made.

The base ratio is given as $2$ tea bags to $3$ spoons of sugar. When we use $5$ tea bags with $b$ spoons of sugar, we can set up the following proportion:

Our goal is to find the value of $P$, which is defined as $2b$. By observing the proportion closely, we can solve directly for $2b$ using cross-multiplication, without needing to find the individual value of $b$ first:

Now we have $P = 15$. We simply compare this with the given value of $Q$, which is $10$.

Thus, we can conclude that the correct relationship is $P$ is greater than $Q$.

Explanation

To determine the selling price of the mixture, we need to know the total capital expenditure and the total weight of the rice obtained. Let's calculate this step-by-step using the concept of weighted average.

The total capital is the sum of the purchase costs for rice type A and type B:

The total weight of the rice after mixing is $20 + 30 = 50$ kg. From this, we can calculate the cost price per kilogram of the mixed rice:

The merchant wants a profit of $10\%$. This means the selling price per kg is the cost price plus the profit, or equivalent to $110\%$ of the cost price.

Thus, the selling price of the mixed rice per kg to achieve a $10\%$ profit is $\text{Rp12,320.00}$.

Explanation

The core of this problem lies in the principle of fair transaction: we only pay for what we actually consume (the soybeans), not the packaging (the sack). In social arithmetic, this concept is known as Net Weight (Netto).

Netto is the clean weight of the goods without packaging. Since the price of soybeans is calculated per kilogram, we must first find the net weight of the soybeans by subtracting the packaging weight (tare) from the gross weight (bruto).

After knowing the actual net weight of the soybeans we are buying, we calculate the total price before the discount.

The entrepreneur receives a discount of $5\%$. This means they only need to pay $95\%$ of the total price. Let's calculate the final amount:

Thus, the money that must be paid by the entrepreneur is $\text{Rp374,300.00}$.

Explanation

This problem can be solved by modeling it into a System of Linear Equations with Three Variables. Let's define the variables first for clarity:

Based on the shopping information of Angga, Bimo, and Cinta, we can set up three equations as follows:

Our target is to find out how much Rizal has to pay if he buys one of each item, which means we are looking for the value of $x + y + z$.

Notice Angga's and Cinta's equations. Both have the exact same amount of sauce ($y$) and pepper ($2z$). This is a shortcut to finding the value of $x$ very quickly! We simply subtract Cinta's equation from Angga's equation:

Now we know the price of soy sauce is $\text{Rp4,000.00}$. Let's substitute this value into Bimo's and Cinta's equations to simplify the system:

We get a new system with two variables. From the equation $y + 2z = 8,000$, we can rewrite it as $y = 8,000 - 2z$. Substitute this into the equation $2y + z = 8,500$:

After getting $z = 2,500$, we can find $y$:

Now all unit prices are known: $x = 4,000$ (soy sauce), $y = 3,000$ (sauce), and $z = 2,500$ (pepper).

Thus, the total amount Rizal must pay is:

So, Rizal must pay $\text{Rp9,500.00}$.

Explanation

Let $B$ be Bimo's current age and $A$ be his Father's current age. We will translate the verbal information from the problem into a precise mathematical model.

First, let's analyze the past condition: "Six years ago, Bimo's age was $4$ years younger than one-sixth of his Father's age." This sentence can be written as the following equation:

Second, let's analyze the current condition: "Bimo's current age is $3$ years older than one-eighth of his Father's age." This statement yields the second equation:

Now we have a system of linear equations with two variables. The next step is to simplify the first equation to make it easier to work with. We will eliminate the fractions and group the variables:

Next, we modify the second equation to have a form consistent with equation $(1)$ to facilitate the elimination process:

With two neat linear equations, we can eliminate variable $A$ by subtracting equation $(1)$ from equation $(2)$:

Having found that Bimo is $9$ years old, we substitute the value of $B$ into equation $(1)$ to find the Father's age:

Finally, we calculate the sum of Bimo's and his Father's current ages:

Thus, the sum of their ages is 57 years.

Explanation

We need to find the number of days in April $2024$ where Pak Saiful does not receive shipments from either Supplier B or C.

First, let's identify the time parameters. April has a total of $30$ days.

Next, we determine the delivery dates for each supplier, starting from April $1$:

The days where Pak Saiful receives a shipment from one or both suppliers are the union of sets $B$ and $C$:

The number of days with shipments is $11$ days.

To find the days without shipments, we subtract the number of shipment days from the total days in April:

Thus, there are $19$ days where Pak Saiful receives no shipments from B or C.

Explanation

We will calculate the total weight of vegetables received by Pak Saiful from Supplier B in April and May $2024$, and then find the difference.

In April $2024$, Supplier B delivers vegetables every $4$ days with a weight of $12$ kg per shipment. The delivery schedule starts on April $1$, so the delivery dates are:

There are $8$ deliveries in April. Thus, the total weight is:

Next, for May $2024$, the weight per shipment increases to $20$ kg. April has $30$ days. Since the last delivery was on April $29$, the next delivery ($4$ days later) falls on May $3$. The delivery dates in May are:

There are also $8$ deliveries in May. Thus, the total weight is:

The difference in vegetable weight received between May and April is:

So, the difference in weight is $64$ kg.

Explanation

We will determine the ratio of the number of male to female employees using the concept of combined average (weighted average).

Let:

• $n_L$ = number of male employees
• $n_P$ = number of female employees
• $\bar{x}_L = 320.000$ (average income of males)
• $\bar{x}_P = 285.000$ (average income of females)
• $\bar{x}_{comb} = 300.000$ (combined average income)

The formula for the combined average is:

Substitute the known values into the formula:

To simplify the calculation, divide all numbers by $1.000$:

Cross-multiply the denominator to the left side:

Group like terms:

Find the ratio $n_L : n_P$:

Thus, the ratio of the number of male to female employees is $3:4$.