# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-10

Exercises: Try Out - Set 10: Real exam simulation to sharpen your skills and build confidence.

---

## Exercise 1

### Question

export const metadata = {
  title: "Question 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

import { Graph } from "../graph";

<Graph
  title="Triangle Pattern"
  description="Determine the correct letter to replace the question mark."
  mode="question"
/>

Observe the image above. The correct letter to replace the question mark (?) in the second triangle is...


### Choices

- [ ] $$V$$
- [ ] $$W$$
- [ ] $$X$$
- [x] $$Y$$
- [ ] $$Z$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Question 1",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

import { Graph } from "../graph";

Define the letters according to their alphabetical order:

- <InlineMath math="A = 1" />
- <InlineMath math="B = 2" />
- <InlineMath math="C = 3" />
- ...
- <InlineMath math="Z = 26" />

Thus, we can convert the letters in the triangles into numbers:

<Graph mode="explanation-numbers" title="Conversion to Numbers" description="Letters are converted to numbers based on alphabetical order." />

Observe that the first triangle follows a pattern:

<MathContainer>
  <BlockMath math="(\text{Left} + \text{Right}) \times (\text{Top} + \text{Bottom}) = \text{Center}" />
  <BlockMath math="(4 + 6) \times (1 + 1) = 10 \times 2 = 20" />
</MathContainer>

The number <InlineMath math="20" /> corresponds to the letter <InlineMath math="T" /> (the <InlineMath math="20" />th letter).

Using the same pattern, the number in the second triangle is:

<MathContainer>
  <BlockMath math="(1 + 4) \times (2 + 3) = 5 \times 5 = 25" />
</MathContainer>

<Graph mode="explanation-result" title="Pattern Calculation" description={<>The calculated result for the second triangle is <InlineMath math="25" />.</>} />

Substitute the number back to the letter sequence. The <InlineMath math="25" />th letter is <InlineMath math="Y" />.

<Graph mode="explanation-final" title="Final Result" description={<>Number <InlineMath math="25" /> is converted back to letter <InlineMath math="Y" />.</>} />

Therefore, the answer is <InlineMath math="Y" />.


---

## Exercise 2

### Question

export const metadata = {
  title: "Question 2",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

import { Graph } from "../graph";

<Graph
  title="Intersecting Angles"
  description={<>Determine the measure of angle <InlineMath math="z" /> in terms of <InlineMath math="x" />.</>}
/>

Based on the figure above, if <InlineMath math="x^\circ = y^\circ" />, what is the measure of angle <InlineMath math="z" /> in terms of <InlineMath math="x" />?


### Choices

- [ ] $$90 - x$$
- [ ] $$90 - 2x$$
- [ ] $$180 - x$$
- [x] $$180 - 2x$$
- [ ] Unknown

### Answer & Explanation

export const metadata = {
  title: "Solution for Question 2",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

import { Graph } from "../graph";

<Graph
  id="question-2"
  title="Angle Analysis"
  description="Relationship between intersecting angles."
/>

From the figure, we can see that the angle vertically opposite to <InlineMath math="z^\circ" /> is the angle between the two diagonal lines at the top. The measure of that angle is equal to <InlineMath math="z^\circ" />.

Since the angles on a straight line sum to <InlineMath math="180^\circ" />, the sum of angle <InlineMath math="x^\circ" />, vertical angle <InlineMath math="z^\circ" />, and <InlineMath math="y^\circ" /> is <InlineMath math="180^\circ" />:

<MathContainer>
  <BlockMath math="x^\circ + z^\circ + y^\circ = 180^\circ" />
</MathContainer>

Given that <InlineMath math="x^\circ = y^\circ" />, we can substitute <InlineMath math="y" /> with <InlineMath math="x" />:

<MathContainer>
  <BlockMath math="x^\circ + z^\circ + x^\circ = 180^\circ" />
  <BlockMath math="z^\circ + 2x^\circ = 180^\circ" />
  <BlockMath math="z^\circ = 180^\circ - 2x^\circ" />
</MathContainer>

Therefore, the measure of angle <InlineMath math="z" /> is <InlineMath math="180 - 2x" />.


---

## Exercise 3

### Question

export const metadata = {
  title: "Question 3",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

At an Independence Day event attended by <InlineMath math="3000" /> people, the committee provides <InlineMath math="20" /> prizes to be drawn during the event. What is the probability that a participant gets a prize?


### Choices

- [ ] $$0.00033$$
- [ ] $$0.00067$$
- [ ] $$0.0033$$
- [x] $$0.0067$$
- [ ] $$0.033$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Question 3",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let <InlineMath math="P(A)" /> be the probability that each person gets a prize, then:

<MathContainer>
  <BlockMath math="P(A) = \frac{\text{number of prizes}}{\text{number of participants}}" />
  <BlockMath math="P(A) = \frac{20}{3000}" />
  <BlockMath math="P(A) = \frac{1}{150}" />
  <BlockMath math="P(A) \approx 0.0067" />
</MathContainer>

Therefore, the probability that a participant gets a prize is <InlineMath math="0.0067" />.


---

## Exercise 4

### Question

export const metadata = {
  title: "Set Operations on Numbers",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Set <InlineMath math="A" /> contains all negative integers, set <InlineMath math="B" /> contains all positive even integers, and set <InlineMath math="C" /> contains all positive integers divisible by <InlineMath math="3" />.

Which of the following numbers is NOT included in the product of two members of sets <InlineMath math="A" />, <InlineMath math="B" />, or <InlineMath math="C" />?


### Choices

- [ ] $$-14$$
- [ ] $$-15$$
- [ ] $$-18$$
- [x] $$14$$
- [ ] $$18$$

### Answer & Explanation

export const metadata = {
    title: "Set Operations Explanation",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "12/27/2025",
};

Let's define the members of each set:

<MathContainer>
  <BlockMath math="A = \{-1, -2, -3, \dots\}" />
  <BlockMath math="B = \{2, 4, 6, \dots\}" />
  <BlockMath math="C = \{3, 6, 9, \dots\}" />
</MathContainer>

We will check each option to see if the number can be formed by multiplying two members of these sets.

#### Analysis of Options

1.  **Option A**: <InlineMath math="-14" />

    <MathContainer>
       <BlockMath math="-14 = (-7) \times 2" />
    </MathContainer>

    Here, <InlineMath math="-7 \in A" /> and <InlineMath math="2 \in B" />. So, <InlineMath math="-14" /> is possible.

2.  **Option B**: <InlineMath math="-15" />

    <MathContainer>
       <BlockMath math="-15 = (-5) \times 3" />
    </MathContainer>

    Here, <InlineMath math="-5 \in A" /> and <InlineMath math="3 \in C" />. So, <InlineMath math="-15" /> is possible.

3.  **Option C**: <InlineMath math="-18" />

    <MathContainer>
       <BlockMath math="-18 = (-3) \times 6" />
    </MathContainer>

    Here, <InlineMath math="-3 \in A" /> and <InlineMath math="6 \in B" /> (also <InlineMath math="6 \in C" />). So, <InlineMath math="-18" /> is possible.

4.  **Option D**: <InlineMath math="14" />

    <MathContainer>
       <BlockMath math="14 = 2 \times 7" />
    </MathContainer>

    Here, <InlineMath math="2 \in B" />. However, <InlineMath math="7" /> is a positive odd integer that is not divisible by <InlineMath math="3" />. This means:
    -   <InlineMath math="7 \notin A" /> (since <InlineMath math="A" /> contains negative integers).
    -   <InlineMath math="7 \notin B" /> (since <InlineMath math="7" /> is odd).
    -   <InlineMath math="7 \notin C" /> (since <InlineMath math="7" /> is not divisible by <InlineMath math="3" />).

    Since one of the factors (<InlineMath math="7" />) is not present in sets <InlineMath math="A" />, <InlineMath math="B" />, or <InlineMath math="C" />, <InlineMath math="14" /> cannot be formed (assuming the pattern involves factors from different sets or positive factors).

5.  **Option E**: <InlineMath math="18" />

    <MathContainer>
       <BlockMath math="18 = 2 \times 9" />
    </MathContainer>

    Here, <InlineMath math="2 \in B" /> and <InlineMath math="9 \in C" />. So, <InlineMath math="18" /> is possible.

Therefore, the number that is NOT included in the product is <InlineMath math="14" />.


---

## Exercise 5

### Question

export const metadata = {
  title: "Maximum Value of Objective Function",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

If the maximum value of <InlineMath math="x + y" /> on the set <InlineMath math="\{(x,y)|x \ge 0, y \ge 0, x + 5y \le 10, 5x + y \le a^2 - 5\}" /> is <InlineMath math="5" />, then the value of <InlineMath math="a = \dots" />


### Choices

- [ ] $$3$$
- [ ] $$4$$
- [x] $$5$$
- [ ] $$16$$
- [ ] $$25$$

### Answer & Explanation

export const metadata = {
  title: "Maximum Value Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given the system of inequalities:

<MathContainer>
  <BlockMath math="\begin{cases} x \ge 0 \\ y \ge 0 \\ x + 5y \le 10 \\ 5x + y \le a^2 - 5 \end{cases}" />
</MathContainer>

We can sum the two main inequalities to find the bound for <InlineMath math="x + y" />:

<MathContainer>
  <BlockMath math="(x + 5y) + (5x + y) \le 10 + (a^2 - 5)" />
  <BlockMath math="6x + 6y \le 5 + a^2" />
  <BlockMath math="6(x + y) \le 5 + a^2" />
</MathContainer>

It is given that the maximum value of <InlineMath math="x + y" /> is <InlineMath math="5" />. Therefore, we substitute this maximum value into the inequality:

<MathContainer>
  <BlockMath math="6(5) = 5 + a^2" />
  <BlockMath math="30 = 5 + a^2" />
  <BlockMath math="a^2 = 25" />
</MathContainer>

Thus, the value of <InlineMath math="a" /> is:

<MathContainer>
  <BlockMath math="a = 5" />
</MathContainer>

(We take the positive value since the options are positive).

So, the value of <InlineMath math="a" /> is <InlineMath math="5" />.


---

## Exercise 6

### Question

export const metadata = {
  title: "Book Discount",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

The price of a math book at ABC store is <InlineMath math="85,000" /> rupiah and a biology book is priced at <InlineMath math="40,000" /> rupiah. Because the store is having an anniversary, all books are given an <InlineMath math="x\%" /> discount. If on that day the price of a math book is <InlineMath math="28,900" /> rupiah. What is the price of the biology book?


### Choices

- [ ] $$11,760$$
- [ ] $$12,000$$
- [x] $$13,600$$
- [ ] $$14,000$$
- [ ] $$15,600$$

### Answer & Explanation

export const metadata = {
  title: "Book Discount Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given:
-   Original price of math book = <InlineMath math="85,000" />
-   Price of math book after discount = <InlineMath math="28,900" />
-   Original price of biology book = <InlineMath math="40,000" />

We need to find the percentage of the price to be paid (price after discount) relative to the original price. Let's find this ratio.

Based on the math book price:

<MathContainer>
  <BlockMath math="\frac{\text{Final Price}}{\text{Original Price}} = \frac{28,900}{85,000}" />
</MathContainer>

Simplify the fraction:

<MathContainer>
  <BlockMath math="\frac{28,900}{85,000} = \frac{289}{850} = \frac{17 \times 17}{17 \times 50} = \frac{17}{50}" />
</MathContainer>

Or as a percentage:

<MathContainer>
  <BlockMath math="\frac{17}{50} \times 100\% = 34\%" />
</MathContainer>

This means the price on that day is <InlineMath math="34\%" /> of the original price.

Now, we calculate the price of the biology book using the same percentage:

<MathContainer>
  <BlockMath math="\text{Biology Price} = 34\% \times 40,000" />
  <BlockMath math="\text{Biology Price} = \frac{34}{100} \times 40,000" />
  <BlockMath math="\text{Biology Price} = 34 \times 400" />
  <BlockMath math="\text{Biology Price} = 13,600" />
</MathContainer>

So, the price of the biology book on that day is <InlineMath math="13,600" /> rupiah.


---

## Exercise 7

### Question

export const metadata = {
  title: "Maximum Value of Quadratic Function",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

If the quadratic function <InlineMath math="y = kx^2 + 8x + (k-1)" /> has an axis of symmetry <InlineMath math="x = 4" />, then the maximum value of the function is...


### Choices

- [ ] $$-14$$
- [x] $$14$$
- [ ] $$-8$$
- [ ] $$8$$
- [ ] $$16$$

### Answer & Explanation

export const metadata = {
  title: "Quadratic Function Maximum Value Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given the quadratic function:

<MathContainer>
  <BlockMath math="y = kx^2 + 8x + (k-1)" />
</MathContainer>

The axis of symmetry for a quadratic function <InlineMath math="ax^2 + bx + c" /> is given by the formula <InlineMath math="x = -\frac{b}{2a}" />. In this case, <InlineMath math="a = k" /> and <InlineMath math="b = 8" />.

Given the axis of symmetry <InlineMath math="x = 4" />, then:

<MathContainer>
  <BlockMath math="4 = -\frac{8}{2k}" />
  <BlockMath math="8k = -8" />
  <BlockMath math="k = -1" />
</MathContainer>

After finding the value of <InlineMath math="k" />, we substitute it back into the original function:

<MathContainer>
  <BlockMath math="y = -1x^2 + 8x + (-1 - 1)" />
  <BlockMath math="y = -x^2 + 8x - 2" />
</MathContainer>

To find the maximum value, we substitute the axis of symmetry value <InlineMath math="x = 4" /> into the function (since the optimum value is always at the axis of symmetry):

<MathContainer>
  <BlockMath math="y = -(4^2) + 8(4) - 2" />
  <BlockMath math="y = -16 + 32 - 2" />
  <BlockMath math="y = 16 - 2" />
  <BlockMath math="y = 14" />
</MathContainer>

Thus, the maximum value of the function is <InlineMath math="14" />.


---

## Exercise 8

### Question

export const metadata = {
  title: "Quadratic Inequality",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given:

1.  <InlineMath math="x > 2" />
2.  <InlineMath math="-3 < x < 2" />
3.  <InlineMath math="x < -3" />
4.  <InlineMath math="x > -3" />

If <InlineMath math="x^2 + x - 6 > 0" />, then the inequality is satisfied by...


### Choices

- [x] $$1$$ and $$3$$
- [ ] $$2$$ and $$4$$
- [ ] $$1$$ and $$4$$
- [ ] $$3$$ and $$4$$
- [ ] $$1$$

### Answer & Explanation

import { Graph } from "../graph";

export const metadata = {
  title: "Quadratic Inequality Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

We will solve the inequality:

<MathContainer>
  <BlockMath math="x^2 + x - 6 > 0" />
</MathContainer>

Factor the quadratic equation:

<MathContainer>
  <BlockMath math="(x - 2)(x + 3) > 0" />
</MathContainer>

The zeros (roots) of the equation are:

<MathContainer>
  <BlockMath math="x = 2 \quad \text{or} \quad x = -3" />
</MathContainer>

We test points on the number line:

<Graph
  title="Number Line"
  description="Visualization of the inequality solution region."
/>

-   For <InlineMath math="x < -3" /> (e.g., <InlineMath math="-4" />): <InlineMath math="(-4-2)(-4+3) = (-6)(-1) = 6" /> (Positive)
-   For <InlineMath math="-3 < x < 2" /> (e.g., <InlineMath math="0" />): <InlineMath math="(0-2)(0+3) = (-2)(3) = -6" /> (Negative)
-   For <InlineMath math="x > 2" /> (e.g., <InlineMath math="3" />): <InlineMath math="(3-2)(3+3) = (1)(6) = 6" /> (Positive)

Since the inequality sign is <InlineMath math="> 0" /> (positive), the solution region is:

<MathContainer>
  <BlockMath math="x < -3 \quad \text{or} \quad x > 2" />
</MathContainer>

Based on the given statements:
1.  <InlineMath math="x > 2" /> (Matches)
2.  <InlineMath math="-3 < x < 2" /> (Does not match)
3.  <InlineMath math="x < -3" /> (Matches)
4.  <InlineMath math="x > -3" /> (Does not match)

Thus, the inequality is satisfied by statements <InlineMath math="1" /> and <InlineMath math="3" />.


---

## Exercise 9

### Question

export const metadata = {
  title: "Construction Probability",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Suppose a contractor is going to build an apartment. The probability of a public protest occurring is <InlineMath math="0.7" />. If there is no protest, the probability that the construction finishes on time is <InlineMath math="0.8" />. If there is a protest, the probability that the construction finishes on time is <InlineMath math="0.4" />.

Determine the probability that the construction finishes on time!


### Choices

- [x] $$0.52$$
- [ ] $$0.62$$
- [ ] $$0.65$$
- [ ] $$0.68$$
- [ ] $$0.70$$

### Answer & Explanation

export const metadata = {
  title: "Construction Probability Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Defined:

-   <InlineMath math="A =" /> event construction finishes on time
-   <InlineMath math="A^c =" /> event construction does not finish on time
-   <InlineMath math="B =" /> event a protest occurs
-   <InlineMath math="B^c =" /> event no protest occurs
-   <InlineMath math="A|B =" /> event construction finishes on time given a protest
-   <InlineMath math="A|B^c =" /> event construction finishes on time given no protest

The probability that construction finishes on time is the sum of the probability of a protest multiplied by the probability of finishing on time given a protest, plus the probability of no protest multiplied by the probability of finishing on time given no protest.

<MathContainer>
  <BlockMath math="P(A) = P(B)P(A|B) + P(B^c)P(A|B^c)" />
</MathContainer>

We know that:

<MathContainer>
  <BlockMath math="P(B) = 0.7, \quad P(B^c) = 1 - 0.7 = 0.3" />
</MathContainer>

<MathContainer>
  <BlockMath math="P(A|B^c) = 0.8, \quad P(A|B) = 0.4" />
</MathContainer>

Then:

<MathContainer>
  <BlockMath math="P(A) = (0.7)(0.4) + (0.3)(0.8)" />
</MathContainer>

<MathContainer>
  <BlockMath math="P(A) = 0.28 + 0.24 = 0.52" />
</MathContainer>

Thus, the probability that the construction finishes on time is <InlineMath math="0.52" />.


---

## Exercise 10

### Question

export const metadata = {
  title: "Average Speed",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Nina runs at a constant speed of <InlineMath math="5 \text{ km/h}" />. And she walks at a speed of <InlineMath math="1 \text{ km/h}" />. If Nina travels <InlineMath math="3 \text{ km}" /> by running for half of the total travel time and walking for the rest, what is the total time taken by Nina?


### Choices

- [ ] $$15$$ minutes
- [ ] $$30$$ minutes
- [ ] $$45$$ minutes
- [x] $$60$$ minutes
- [ ] $$75$$ minutes

### Answer & Explanation

export const metadata = {
  title: "Average Speed Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

We know that:

<MathContainer>
  <BlockMath math="\text{speed} \times \text{time} = \text{distance}" />
</MathContainer>

Then:

| | Speed | Time | Distance |
| :--- | :--- | :--- | :--- |
| Run | <InlineMath math="5 \text{ km/h}" /> | <InlineMath math="x \text{ hours}" /> | <InlineMath math="5x \text{ km}" /> |
| Walk | <InlineMath math="1 \text{ km/h}" /> | <InlineMath math="x \text{ hours}" /> | <InlineMath math="x \text{ km}" /> |

Nina travels a distance of <InlineMath math="3 \text{ km}" />.

<MathContainer>
  <BlockMath math="\text{running} + \text{walking} = 3 \text{ km}" />
</MathContainer>

<MathContainer>
  <BlockMath math="5x \text{ km/h} + x \text{ km/h} = 3 \text{ km}" />
</MathContainer>

<MathContainer>
  <BlockMath math="6x \frac{\text{km}}{\text{h}} = 3 \text{ km}" />
</MathContainer>

<MathContainer>
  <BlockMath math="x = \frac{1}{2} \text{ hour}" />
</MathContainer>

The total time taken by Nina is the sum of the time spent running and walking, which is:

<MathContainer>
  <BlockMath math="\text{Total time} = x + x = 2x" />
</MathContainer>

<MathContainer>
  <BlockMath math="\text{Total time} = 2 \left(\frac{1}{2}\right) = 1 \text{ hour}" />
</MathContainer>

Convert to minutes:

<MathContainer>
  <BlockMath math="1 \text{ hour} = 60 \text{ minutes}" />
</MathContainer>

Thus, the total time taken by Nina is <InlineMath math="60" /> minutes.


---

## Exercise 11

### Question

export const metadata = {
  title: "Putri and Tina's Age",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Putri and Tina are celebrating their birthdays today. Currently, Tina's age is <InlineMath math="2" /> times Putri's age. If in <InlineMath math="6" /> years, Putri's age is <InlineMath math="\frac{2}{3}" /> of Tina's age, what is Putri's current age?


### Choices

- [ ] $$5$$
- [x] $$6$$
- [ ] $$7$$
- [ ] $$8$$
- [ ] $$9$$

### Answer & Explanation

export const metadata = {
  title: "Putri and Tina's Age Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let Putri's current age be <InlineMath math="x" />, then Tina's age is <InlineMath math="2x" />.

In <InlineMath math="6" /> years:

<MathContainer>
  <BlockMath math="\text{Putri's Age} = x + 6" />
  <BlockMath math="\text{Tina's Age} = 2x + 6" />
</MathContainer>

Given that in <InlineMath math="6" /> years Putri's age will be <InlineMath math="\frac{2}{3}" /> of Tina's age, then:

<MathContainer>
  <BlockMath math="x + 6 = \frac{2}{3}(2x + 6)" />
</MathContainer>

<MathContainer>
  <BlockMath math="3(x + 6) = 2(2x + 6)" />
</MathContainer>

<MathContainer>
  <BlockMath math="3x + 18 = 4x + 12" />
</MathContainer>

<MathContainer>
  <BlockMath math="18 - 12 = 4x - 3x" />
</MathContainer>

<MathContainer>
  <BlockMath math="6 = x" />
</MathContainer>

Thus, Putri's current age is <InlineMath math="6" /> years.


---

## Exercise 12

### Question

export const metadata = {
  title: "House Construction Completion",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

To build a house, a contractor needs <InlineMath math="33" /> workers for <InlineMath math="21" /> days. If only <InlineMath math="14" /> workers arrive, how long will they take to complete the job?


### Choices

- [ ] $$48$$ days
- [ ] $$48.5$$ days
- [ ] $$49$$ days
- [x] $$49.5$$ days
- [ ] $$50$$ days

### Answer & Explanation

export const metadata = {
  title: "House Construction Completion Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given:

<MathContainer>
  <BlockMath math="33 \text{ workers} \rightarrow 21 \text{ days}" />
  <BlockMath math="14 \text{ workers} \rightarrow x \text{ days}" />
</MathContainer>

Using inverse proportion:

<MathContainer>
  <BlockMath math="\frac{33}{14} = \frac{x}{21}" />
</MathContainer>

<MathContainer>
  <BlockMath math="14x = 33 \times 21" />
</MathContainer>

<MathContainer>
  <BlockMath math="14x = 693" />
</MathContainer>

<MathContainer>
  <BlockMath math="x = \frac{693}{14}" />
</MathContainer>

<MathContainer>
  <BlockMath math="x = 49.5" />
</MathContainer>

Thus, they can complete the job in <InlineMath math="49.5" /> days.


---

## Exercise 13

### Question

export const metadata = {
  title: "Ordering Numbers",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

The correct order of the numbers below from largest to smallest is...

<MathContainer>
  <BlockMath math="\frac{3}{4}; 0.875; 1\frac{1}{8}; 12.5\%; \frac{1}{\sqrt{2}}" />
</MathContainer>


### Choices

- [x] $$1\frac{1}{8}; 0.875; \frac{3}{4}; \frac{1}{\sqrt{2}}; 12.5\%$$
- [ ] $$1\frac{1}{8}; \frac{3}{4}; \frac{1}{\sqrt{2}}; 12.5\%; 0.875$$
- [ ] $$1\frac{1}{8}; 0.875; 12.5\%; \frac{3}{4}; \frac{1}{\sqrt{2}}$$
- [ ] $$0.875; 1\frac{1}{8}; \frac{3}{4}; \frac{1}{\sqrt{2}}; 12.5\%$$
- [ ] $$\frac{1}{\sqrt{2}}; 1\frac{1}{8}; 0.875; \frac{3}{4}; 12.5\%$$

### Answer & Explanation

export const metadata = {
   title: "Ordering Numbers",
   authors: [{ name: "Nabil Akbarazzima Fatih" }],
   date: "12/27/2025",
};

To order these numbers, we can convert all of them into decimal form for easier comparison.

<MathContainer>
  <BlockMath math="1\frac{1}{8} = 1 + 0.125 = 1.125" />
  <BlockMath math="0.875 = 0.875" />
  <BlockMath math="\frac{3}{4} = 0.75" />
  <BlockMath math="\frac{1}{\sqrt{2}} = \frac{1}{1.414} \approx 0.707" />
  <BlockMath math="12.5\% = \frac{12.5}{100} = 0.125" />
</MathContainer>

Now we order them from largest to smallest:

<MathContainer>
  <BlockMath math="1.125 > 0.875 > 0.75 > 0.707 > 0.125" />
</MathContainer>

So the correct order is:

<MathContainer>
  <BlockMath math="1\frac{1}{8}; 0.875; \frac{3}{4}; \frac{1}{\sqrt{2}}; 12.5\%" />
</MathContainer>


---

## Exercise 14

### Question

export const metadata = {
  title: "Recurring Decimal Numbers",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

The recurring decimal number <InlineMath math="1.5454545454..." /> in common fraction form is...


### Choices

- [ ] $$\frac{10}{16}$$
- [ ] $$\frac{11}{17}$$
- [ ] $$\frac{9}{10}$$
- [ ] $$\frac{27}{26}$$
- [x] $$\frac{17}{11}$$

### Answer & Explanation

export const metadata = {
  title: "Recurring Decimal Numbers",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let <InlineMath math="a = 1.5454545454..." />

Multiply <InlineMath math="a" /> by <InlineMath math="100" />, so:

<MathContainer>
  <BlockMath math="100a = 154.54545454..." />
</MathContainer>

Subtract <InlineMath math="a" /> from <InlineMath math="100a" />, so:

<MathContainer>
  <BlockMath math="100a - a = 154.54545454... - 1.5454545454..." />
  <BlockMath math="99a = 153" />
  <BlockMath math="a = \frac{153}{99} = \frac{17}{11}" />
</MathContainer>

Therefore, the fraction form is <InlineMath math="\frac{17}{11}" />.


---

## Exercise 15

### Question

export const metadata = {
  title: "Number Sequence",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

<InlineMath math="4, 5, 8, 2, \dots, \dots, 32, -4, 64, -7" />

The correct values to complete the number sequence are...


### Choices

- [ ] $$12, -1$$
- [x] $$16, -1$$
- [ ] $$-2, 12$$
- [ ] $$12, 16$$
- [ ] $$-1, -2$$

### Answer & Explanation

export const metadata = {
  title: "Number Sequence",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

This sequence consists of two alternating patterns.

First pattern (odd positions):

<MathContainer>
  <BlockMath math="4 \xrightarrow{\times 2} 8 \xrightarrow{\times 2} \mathbf{16} \xrightarrow{\times 2} 32 \xrightarrow{\times 2} 64" />
</MathContainer>

Second pattern (even positions):

<MathContainer>
  <BlockMath math="5 \xrightarrow{-3} 2 \xrightarrow{-3} \mathbf{-1} \xrightarrow{-3} -4 \xrightarrow{-3} -7" />
</MathContainer>

The missing terms are the <InlineMath math="5" />th term (from the first pattern) and the <InlineMath math="6" />th term (from the second pattern).

For the <InlineMath math="5" />th term:

<MathContainer>
  <BlockMath math="8 \times 2 = 16" />
</MathContainer>

For the <InlineMath math="6" />th term:

<MathContainer>
  <BlockMath math="2 - 3 = -1" />
</MathContainer>

Therefore, the correct values to complete the sequence are <InlineMath math="16" /> and <InlineMath math="-1" />.


---

## Exercise 16

### Question

export const metadata = {
  title: "Venn Diagram",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

In a class of <InlineMath math="50" /> students, each student has at least one favorite subject among three subjects: Mathematics, Biology, and Physics.

The data is as follows:
- <InlineMath math="35" /> students like Mathematics
- <InlineMath math="20" /> students like Biology
- <InlineMath math="15" /> students like Physics
- <InlineMath math="5" /> students like Physics and Biology
- <InlineMath math="13" /> students like Mathematics and Biology
- <InlineMath math="7" /> students like Mathematics and Physics

How many students like Mathematics, Biology, and Physics?


### Choices

- [ ] $$1$$
- [ ] $$2$$
- [ ] $$3$$
- [ ] $$4$$
- [x] $$5$$

### Answer & Explanation

export const metadata = {
  title: "Venn Diagram",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let <InlineMath math="x" /> be the number of students who like all three subjects (Mathematics, Physics, and Biology).

We can use the inclusion-exclusion principle for sets:

<MathContainer>
  <BlockMath math="\begin{aligned} n(M \cup B \cup P) &= n(M) + n(B) + n(P) \\ &- (n(M \cap B) + n(M \cap P) + n(B \cap P)) \\ &+ n(M \cap B \cap P) \end{aligned}" />
</MathContainer>

Given:

<MathContainer>
  <BlockMath math="n(M \cup B \cup P) = 50" />
  <BlockMath math="n(M) = 35" />
  <BlockMath math="n(B) = 20" />
  <BlockMath math="n(P) = 15" />
  <BlockMath math="n(M \cap B) = 13" />
  <BlockMath math="n(M \cap P) = 7" />
  <BlockMath math="n(B \cap P) = 5" />
  <BlockMath math="n(M \cap B \cap P) = x" />
</MathContainer>

Substitute these values into the formula:

<MathContainer>
  <BlockMath math="50 = 35 + 20 + 15 - (13 + 7 + 5) + x" />
  <BlockMath math="50 = 70 - 25 + x" />
  <BlockMath math="50 = 45 + x" />
  <BlockMath math="x = 5" />
</MathContainer>

Therefore, the number of students who like all three subjects is <InlineMath math="5" />.


---

## Exercise 17

### Question

export const metadata = {
  title: "Broken Egg Probability",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

A basket contains <InlineMath math="10" /> duck eggs and <InlineMath math="20" /> chicken eggs. Half of the duck eggs are broken, and one-fourth of the chicken eggs are broken.

If one egg is taken from the basket, what is the probability that the taken egg is a duck egg or a broken egg?


### Choices

- [ ] $$\frac{1}{3}$$
- [ ] $$\frac{2}{3}$$
- [x] $$0.5$$
- [ ] $$0.333$$
- [ ] $$\frac{1}{4}$$

### Answer & Explanation

export const metadata = {
  title: "Egg Probability Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Let's define the events as follows:
- <InlineMath math="S" /> = Sample space (total eggs)
- <InlineMath math="D" /> = Event of picking a duck egg
- <InlineMath math="C" /> = Event of picking a chicken egg
- <InlineMath math="B" /> = Event of picking a broken egg

First, we calculate the total number of eggs in the basket:

<MathContainer>
  <BlockMath math="n(S) = n(D) + n(C) = 10 + 20 = 30" />
</MathContainer>

Next, we calculate the number of broken eggs:
- Broken duck eggs: <InlineMath math="\frac{1}{2} \times 10 = 5" />
- Broken chicken eggs: <InlineMath math="\frac{1}{4} \times 20 = 5" />

The total number of broken eggs (<InlineMath math="n(B)" />) is:

<MathContainer>
  <BlockMath math="n(B) = 5 + 5 = 10" />
</MathContainer>

We are asked to find the probability of picking a duck egg **or** a broken egg (<InlineMath math="D \cup B" />). The formula for the probability of the union of two events is:

<MathContainer>
  <BlockMath math="P(D \cup B) = \frac{n(D \cup B)}{n(S)}" />
</MathContainer>

Where <InlineMath math="n(D \cup B)" /> can be calculated using the inclusion-exclusion principle:

<MathContainer>
  <BlockMath math="n(D \cup B) = n(D) + n(B) - n(D \cap B)" />
</MathContainer>

- <InlineMath math="n(D) = 10" /> (number of duck eggs)
- <InlineMath math="n(B) = 10" /> (total number of broken eggs)
- <InlineMath math="n(D \cap B)" /> is the number of eggs that are both duck eggs **and** broken. From the calculation above, there are <InlineMath math="5" /> broken duck eggs.

Thus:

<MathContainer>
  <BlockMath math="n(D \cup B) = 10 + 10 - 5 = 15" />
</MathContainer>

So the probability is:

<MathContainer>
  <BlockMath math="P(D \cup B) = \frac{15}{30} = \frac{1}{2} = 0.5" />
</MathContainer>

Therefore, the probability of picking a duck egg or a broken egg is <InlineMath math="0.5" />.


---

## Exercise 18

### Question

export const metadata = {
  title: "Intersection of Two Sets",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

Given two sets:

<MathContainer>
  <BlockMath math="A = \{x | -3 < x < 8, x \in \text{integers}\}" />
  <BlockMath math="B = \{x | x \leq 7, x \in \text{integers}\}" />
</MathContainer>

The intersection of these two sets is...


### Choices

- [ ] $$\{x | -3 \leq x, x \in \text{integers}\}$$
- [ ] $$\{x | -3 < x, x \in \text{integers}\}$$
- [ ] $$\{x | x \leq 7, x \in \text{integers}\}$$
- [x] $$\{x | -3 < x \leq 7, x \in \text{integers}\}$$
- [ ] $$\{x | -3 < x < 7, x \in \text{integers}\}$$

### Answer & Explanation

export const metadata = {
  title: "Set Intersection Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

First, let's list the members of set <InlineMath math="A" />. Set <InlineMath math="A" /> contains integers between <InlineMath math="-3" /> and <InlineMath math="8" /> (exclusive):

<MathContainer>
  <BlockMath math="A = \{-2, -1, 0, 1, 2, 3, 4, 5, 6, 7\}" />
</MathContainer>

Next, set <InlineMath math="B" /> contains all integers less than or equal to <InlineMath math="7" />:

<MathContainer>
  <BlockMath math="B = \{..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7\}" />
</MathContainer>

The intersection of two sets (<InlineMath math="A \cap B" />) is the set containing elements that are present in both sets.

Since all members of set <InlineMath math="A" /> (from <InlineMath math="-2" /> to <InlineMath math="7" />) are also contained in set <InlineMath math="B" />, the intersection is set <InlineMath math="A" /> itself:

<MathContainer>
  <BlockMath math="A \cap B = \{-2, -1, 0, 1, 2, 3, 4, 5, 6, 7\}" />
</MathContainer>

Converting this back to set-builder notation, the elements are integers greater than <InlineMath math="-3" /> and less than or equal to <InlineMath math="7" />:

<MathContainer>
  <BlockMath math="A \cap B = \{x | -3 < x \leq 7, x \in \text{integers}\}" />
</MathContainer>


---

## Exercise 19

### Question

export const metadata = {
  title: "Average Score Increase",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

In Semester <InlineMath math="I" />, Hani took <InlineMath math="5" /> Mathematics exams and obtained an average score of <InlineMath math="7.5" />. In one academic year, Hani took a total of <InlineMath math="11" /> mathematics exams and obtained an average score of <InlineMath math="8.0" />.

What is the increase in Hani's average mathematics exam score from Semester <InlineMath math="I" /> to Semester <InlineMath math="II" />?


### Choices

- [x] $$0.92$$
- [ ] $$0.82$$
- [ ] $$0.7$$
- [ ] $$1.2$$
- [ ] $$0.96$$

### Answer & Explanation

export const metadata = {
  title: "Average Increase Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

First, let's calculate the total exam score for Semester <InlineMath math="I" />:

<MathContainer>
  <BlockMath math="\text{Total Score Semester } I = 5 \times 7.5 = 37.5" />
</MathContainer>

Next, we calculate the total exam score for the whole year (Semester <InlineMath math="I" /> & <InlineMath math="II" />):

<MathContainer>
  <BlockMath math="\text{Total Score 1 Year} = 11 \times 8 = 88" />
</MathContainer>

To find the total exam score for Semester <InlineMath math="II" />, we subtract the Semester <InlineMath math="I" /> total from the yearly total:

<MathContainer>
  <BlockMath math="\text{Total Score Semester } II = 88 - 37.5 = 50.5" />
</MathContainer>

The number of exams in Semester <InlineMath math="II" /> is the difference between the total exams and Semester <InlineMath math="I" /> exams:

<MathContainer>
  <BlockMath math="\text{Number of Exams Semester } II = 11 - 5 = 6" />
</MathContainer>

Now we can calculate the average score for Semester <InlineMath math="II" />:

<MathContainer>
  <BlockMath math="\text{Average Semester } II = \frac{50.5}{6} \approx 8.4166... \approx 8.42" />
</MathContainer>

Finally, we calculate the increase in the average score from Semester <InlineMath math="I" /> to Semester <InlineMath math="II" />:

<MathContainer>
  <BlockMath math="\text{Increase} = \text{Average Semester } II - \text{Average Semester } I" />
  <BlockMath math="\text{Increase} = 8.42 - 7.5 = 0.92" />
</MathContainer>

Therefore, the increase in Hani's mathematics exam average is <InlineMath math="0.92" />.


---

## Exercise 20

### Question

export const metadata = {
  title: "Relationship between x and y",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

The relationship between the values of <InlineMath math="x" /> and <InlineMath math="y" /> from the following equations is...

<MathContainer>
  <BlockMath math="4x - 16 = 24" />
  <BlockMath math="2x + y - 16 = 12" />
</MathContainer>


### Choices

- [ ] $$x < y$$
- [x] $$x > y$$
- [ ] $$x = y$$
- [ ] $$x = -y$$
- [ ] $$x + y = 1$$

### Answer & Explanation

export const metadata = {
  title: "x and y Value Explanation",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/27/2025",
};

First, we find the value of <InlineMath math="x" /> from the first equation:

<MathContainer>
  <BlockMath math="4x - 16 = 24" />
  <BlockMath math="4x = 24 + 16" />
  <BlockMath math="4x = 40" />
  <BlockMath math="x = \frac{40}{4} = 10" />
</MathContainer>

After finding that <InlineMath math="x = 10" />, we substitute it into the second equation to find the value of <InlineMath math="y" />:

<MathContainer>
  <BlockMath math="2x + y - 16 = 12" />
  <BlockMath math="2(10) + y - 16 = 12" />
  <BlockMath math="20 + y - 16 = 12" />
  <BlockMath math="y + 4 = 12" />
  <BlockMath math="y = 12 - 4" />
  <BlockMath math="y = 8" />
</MathContainer>

Finally, we compare the values of <InlineMath math="x" /> and <InlineMath math="y" />:

<MathContainer>
  <BlockMath math="x = 10" />
  <BlockMath math="y = 8" />
</MathContainer>

Since <InlineMath math="10 > 8" />, then:

<MathContainer>
  <BlockMath math="x > y" />
</MathContainer>


---
