CTJan27 Online Year 8 Mathematical Reasoning 27012026

1

Solve the compound inequality and graph the solution set: $2x - 5 > 7 \text{ AND } -4x + 1 \geq 13$.

$x < -3 \text{ or } x > 6$ $-3 \leq x \leq 6$ $\emptyset$ (No Solution) $x \geq -3 \text{ and } x \leq 6$

2

Which inequality represents the solution set $x < -2 \text{ or } x \geq 4$?

$-10 < 5x \text{ and } -x \leq -4$ $3x + 7 < 1 \text{ or } -2x + 5 \leq -3$ $|x - 1| > 3$ $3x - 1 < -7 \text{ or } 2x - 1 \geq 7$

3

Factor the trinomial completely: $6x^2 - 19x + 10$.

$(6x - 2)(x - 5)$ $(3x + 2)(2x + 5)$ $(3x - 2)(2x - 5)$ $(6x - 5)(x - 2)$

4

Factor completely: $49x^4 - 81y^2$.

$(7x^2 - 9y)(7x^2 + 9y)$ $(7x^2 - 9y)^2$ $(7x - 9y)(7x + 9y)(x^2)$ $(7x^2 - 81y)(7x^2 + 1)$

5

Use the Quadratic Formula to solve $2x^2 = 5x - 1$.

$x = \frac{5 \pm \sqrt{17}}{4}$ $x = \frac{-5 \pm \sqrt{17}}{4}$ $x = 5 \pm \frac{\sqrt{33}}{4}$ $x = \frac{5 \pm \sqrt{33}}{4}$

6

How many distinct real solutions does the equation $3x^2 - 7x + 5 = 0$ have?

One real solution Two distinct real solutions Zero real solutions Infinitely many solutions

7

A parabola is defined by $y = -0.5(x+1)^2 - 9$. What is the range of the function?

$y \geq -1$ $y \leq -9$ $y \geq -9$ All real numbers

8

Describe the sequence of transformations from the parent function $f(x)=x^2$ to $g(x) = -3(x+2)^2 + 5$.

Reflected across the y-axis, vertically stretched by 3, shifted right 2, shifted up 5. Reflected across the x-axis, vertically compressed by 3, shifted left 2, shifted up 5. Reflected across the x-axis, vertically stretched by 3, shifted left 2, shifted up 5. Reflected across the x-axis, vertically stretched by 3, shifted right 2, shifted down 5.

9

The parabola $y = 2(x-h)^2 + k$ has its vertex on the line $x=5$. If the parabola passes through the point $(3, 11)$, what is the value of $k$?

$k = -1$ $k = 3$ $k = 19$ $k = 8$

10

A total of $250$ tickets were sold for a play. Adult tickets cost $\$10$ each and student tickets cost $\$5$ each. Total revenue was $\$2000$. If the number of adult tickets sold was $50$ more than the number of student tickets, how many student tickets ($S$) were sold?

$S = 75$ $S = 100$ $S = 125$ $S = 150$

11

A company produces gadgets ($G$) and widgets ($W$). Assembly time is limited to $100$ hours, and finishing time is limited to $60$ hours. Gadgets require $2$ hours assembly and $1$ hour finishing. Widgets require $1$ hour assembly and $2$ hours finishing. Which system of inequalities models these production constraints?

$\begin{cases} G + 2W \leq 100 \\ 2G + W \leq 60 \end{cases}$ $\begin{cases} 2G + W \leq 100 \\ G + 2W \leq 60 \end{cases}$ $\begin{cases} 2G + 2W \leq 160 \\ G + W \leq 100 \end{cases}$ $\begin{cases} 2G + W \geq 100 \\ G + 2W \geq 60 \end{cases}$

12

A product was initially marked up by $50\%$. Later, it was sold during a promotional sale offering a $20\%$ discount off the marked price. If the final sale price was $\$180$, what was the original cost of the product?

$\$150$ $\$144$ $\$130$ $\$187.50$

13

A store applied a $20\%$ discount and then added $10\%$ sales tax to the discounted price. If the final price paid by the customer was $\$88.00$, what was the original price before the discount?

$\$100.00$ $\$96.80$ $\$98.20$ $\$110.00$

14

An investment of $\$5000$ grew to $\$6050$ after 2 years, compounded annually. What was the annual percentage growth rate?

$10\%$ $10.5\%$ $12\%$ $21\%$

15

A rectangular pool has a capacity of $600 \text{ m}^3$. If the pool is currently $\frac{2}{5}$ full, and water is removed until it is only $\frac{1}{6}$ full, what volume of water, in cubic meters, was removed?

$100 \text{ m}^3$ $140 \text{ m}^3$ $240 \text{ m}^3$ $360 \text{ m}^3$

16

A rectangular garden has an area of $48 \text{ sq. ft}$. The width of the garden is $\frac{3}{4}$ of its length. What is the perimeter of the garden?

$24 \text{ ft}$ $28 \text{ ft}$ $32 \text{ ft}$ $40 \text{ ft}$

17

If $y$ varies directly as the cube of $x$, and $y=5$ when $x=2$, find the value of $y$ when $x=4$.

$y = 20$ $y = 40$ $y = 10$ $y = 32$

18

The time $T$ (in hours) required to complete a job varies inversely with the number of workers $W$. If 5 workers can complete the job in 12 hours, how long will it take 8 workers to complete the same job?

$7.5 \text{ hours}$ $19.2 \text{ hours}$ $9 \text{ hours}$ $8 \text{ hours}$

19

$Z$ varies jointly as $X$ and the square of $Y$. If $Z=100$ when $X=2$ and $Y=5$, find $Z$ when $X=3$ and $Y=2$.

$Z = 24$ $Z = 30$ $Z = 48$ $Z = 12$

20

A car travels from City A to City B at $60 \text{ mph}$ and returns immediately from B to A at $40 \text{ mph}$. What is the average speed for the entire round trip?

$50 \text{ mph}$ $48 \text{ mph}$ $45 \text{ mph}$ $52 \text{ mph}$

21

A boat travels $20 \text{ miles}$ downstream in $1 \text{ hour}$ and $20 \text{ miles}$ upstream in $2 \text{ hours}$. What is the speed of the current?

$15 \text{ mph}$ $5 \text{ mph}$ $10 \text{ mph}$ $2.5 \text{ mph}$

22

Two cars leave the same spot traveling in the same direction. Car A leaves 1 hour earlier at $50 \text{ mph}$. Car B leaves later at $70 \text{ mph}$. How long after Car B leaves will it catch Car A?

$1.5 \text{ hours}$ $2 \text{ hours}$ $2.5 \text{ hours}$ $3 \text{ hours}$

23

Convert $15 \text{ meters per second}$ to $\text{kilometers per hour}$.

$54 \text{ km/h}$ $45 \text{ km/h}$ $81 \text{ km/h}$ $90 \text{ km/h}$

24

The density of gold is $19.3 \text{ g/cm}^3$. Convert this density to $\text{kg/m}^3$.

$19,300 \text{ kg/m}^3$ $193 \text{ kg/m}^3$ $1.93 \text{ kg/m}^3$ $193,000 \text{ kg/m}^3$

25

Consider datasets A: $\{5, 5, 5, 5\}$ and B: $\{0, 5, 5, 10\}$. Which statement regarding their standard deviations ($\sigma$) is true?

$\sigma_A = \sigma_B$ because they have the same mean. $\sigma_A = 0$ and $\sigma_B > 0$, so $\sigma_A < \sigma_B$. $\sigma_A > \sigma_B$ because Dataset A has less variation. $\sigma_B$ cannot be calculated because the data is non-normally distributed.

26

A dataset is given by $\{8, 12, 15, x, 25\}$. If the mean of the data set is equal to the median, what is the value of $x$? (Assume $x$ is placed such that the median is well defined.)

$x = 10$ $x = 15$ $x = 18$ $x = 20$

27

A dataset $\{10, 15, 15, 20, 25\}$ has a mean of 17 and a median of 15. If the outlier value of $25$ is replaced by $75$, which measure of central tendency changes the most dramatically?

The Median, increasing significantly. The Mode, shifting to a higher value. The Mean, increasing from 17 to 27. Both Mean and Median change equally.

28

The perimeter of a rectangular enclosure must be at least $40 \text{ meters}$. If the length ($L$) is $3 \text{ m}$ more than twice the width ($W$), write the inequality representing the constraint on the width ($W$).

$4W + 3 \geq 40$ $6W + 6 \geq 40$ $3W + 6 \geq 40$ $2W + 3 \leq 40$

29

Translate the following verbal model into an algebraic equation: 'The square root of the quantity that is 5 more than the quotient of $x$ and $y$ is equal to the product of $x$ and the reciprocal of $y$.'

$\sqrt{5 + x/y} = x \cdot y$ $\sqrt{5 + x/y} = x/y$ $5 + x/y = x \cdot (1/y)^2$ $\sqrt{x + 5y} = x/y$

30

A storage tank was $\frac{3}{4}$ full. If $\frac{1}{5}$ of the water remaining was pumped out, what fraction of the total tank capacity remains in the tank?

$\frac{11}{20}$ $\frac{3}{5}$ $\frac{4}{5}$ $\frac{7}{10}$

31

A chemist needs $10 \text{ liters}$ of a $35\%$ acid solution. She has $20\%$ solution and $50\%$ solution available. How many liters of the $50\%$ solution must she use?

$6 \text{ liters}$ $5 \text{ liters}$ $3 \text{ liters}$ $4 \text{ liters}$

32

A patient needs a drug dosage of $0.5 \text{ mg}$ per kilogram of body weight. If the patient weighs $150 \text{ lbs}$, and $1 \text{ kg} \approx 2.2 \text{ lbs}$, what is the required dosage in milligrams? (Round to the nearest tenth.)

$34.1 \text{ mg}$ $75.0 \text{ mg}$ $68.2 \text{ mg}$ $37.5 \text{ mg}$

33

If a parabola has a vertex at $(4, -1)$ and passes through the point $(2, 3)$, what is its equation in vertex form?

$y = -1(x-4)^2 - 1$ $y = 1(x+4)^2 - 1$ $y = (x-4)^2 - 1$ $y = 2(x-4)^2 + 3$

34

A population currently stands at $120,000$ and grows at a rate of $2.5\%$ per year, compounded annually. What will the population be in 4 years? (Round to the nearest whole number.)

$132,000$ $132,458$ $129,000$ $133,500$

35

Factor completely: $12x^2 + 14x - 6$.

$2(6x - 1)(x + 3)$ $2(3x + 1)(2x - 3)$ $2(3x - 1)(2x + 3)$ $(6x - 2)(2x + 3)$

36

What are the $x$-intercepts of the parabola defined by $y = (x-3)^2 - 16$?

$(4, 0)$ and $(-4, 0)$ $(3, 0)$ and $(-1, 0)$ $(7, 0)$ and $(-1, 0)$ $(0, 7)$ and $(0, -1)$

37

If $a$ is inversely proportional to $b$, and $b$ is directly proportional to $c^2$. If $a=10$ when $c=3$, find $a$ when $c=6$.

$a = 5$ $a = 2.5$ $a = 20$ $a = 1.25$

38

If the length of a side of a square is $4x-3$ and its area is $16x^2 - 24x + 9$, what expression represents the perimeter?

$16x - 12$ $8x - 6$ $4x + 3$ $12x - 9$

39

If $x$ is an integer, what is the greatest possible value of $x$ that satisfies the inequality $-10 < 2(x+1) \leq 14$?

$5$ $6$ $7$ $-6$

40

The function $h(t) = -16t^2 + 96t + 112$ models the height of a rocket in feet, $t$ seconds after launch. At what time does the rocket reach its maximum height?

$3 \text{ seconds}$ $4 \text{ seconds}$ $6 \text{ seconds}$ $7 \text{ seconds}$

41

A product initially cost $\$200$. Its price was reduced by $10\%$, and later the reduced price was further reduced by $5\%$. What is the total percent decrease from the original price?

$15.0\%$ $14.5\%$ $15.5\%$ $13.5\%$

42

A recipe calls for $\frac{2}{3} \text{ cup}$ of flour. If a baker needs to make $\frac{5}{2}$ times the recipe amount, and already has $\frac{1}{4} \text{ cup}$ of flour, how much more flour (in cups) is needed?

$\frac{17}{12} \text{ cups}$ $\frac{7}{4} \text{ cups}$ $\frac{5}{3} \text{ cups}$ $\frac{11}{6} \text{ cups}$

43

A train travels $300 \text{ miles}$ at a constant speed. If the speed had been $10 \text{ mph}$ faster, the trip would have taken $1 \text{ hour}$ less. What was the original speed of the train?

$40 \text{ mph}$ $50 \text{ mph}$ $60 \text{ mph}$ $75 \text{ mph}$

44

A cyclist completes a training ride of $20 \text{ miles}$. The first $12 \text{ miles}$ are covered at $15 \text{ mph}$, and the remaining $8 \text{ miles}$ are covered at $10 \text{ mph}$. What is the average speed for the entire ride?

$12.5 \text{ mph}$ $12 \text{ mph}$ $13.33 \text{ mph}$ $11.5 \text{ mph}$

45

A rare type of bacterial culture doubles its biomass every $4$ hours. A scientist begins an experiment with an initial mass $M$. After $T$ hours, the biomass reaches $128M$. If another similar culture starts simultaneously with an initial mass of $4M$, how long, in hours, will it take this second culture to reach the same final mass of $128M$?

$T - 8$ $T - 12$ $T + 4$ $T - 4$

46

A complex financial transaction involves a sequential change in value. An asset initially valued at $V$ first increased by $x\%$ and then decreased by $x\%$. The final value was $0.99 V$. If a second asset, valued at $2V$, was increased by $2x\%$ and then decreased by $2x\%$, what is its final value?

$1.92 V$ $1.98 V$ $1.99 V$ $2.00 V$

47

A team of three robots, $R_A, R_B$, and $R_C$, are assigned a construction task. Alone, $R_A$ takes $6$ hours, $R_B$ takes $8$ hours, and $R_C$ takes $12$ hours to complete the task. They begin working together. However, after $R_A$ works for $1$ hour, it malfunctions and stops. $R_B$ and $R_C$ continue until the task is complete. How many total hours elapsed from start to finish?

$\frac{16}{5}$ hours $4.23$ hours $4$ hours $\frac{13}{4}$ hours

48

The 'Aethel-Clock' is an unusual timekeeper where a full cycle consists of $17$ hours. If the clock currently reads $11$ Aethel-hours, what will the time read exactly $300$ Aethel-hours from now?

$10$ Aethel-hours $12$ Aethel-hours $5$ Aethel-hours $6$ Aethel-hours

49

A rectangular plot of land measures $150$ meters by $120$ meters. A central portion, $30$ meters by $30$ meters, is reserved for a sculpture garden and cannot be tiled. The remaining area is to be tiled using square tiles of size $2$ meters by $2$ meters. Assuming tiles must be laid edge-to-edge without cutting, what is the maximum number of tiles that can be fully placed?

$4,275$ $4,491$ $4,500$ $4,505$

50

A manufacturer mixes two different acid solutions. Solution A is $20\%$ acid by volume, and Solution B is $75\%$ acid by volume. The manufacturer mixes $V_A$ liters of Solution A with $V_B$ liters of Solution B to create a $40\%$ acid solution. If $10$ liters of pure water are added to this resultant mixture, the final concentration drops to $30\%$. What is the initial total volume, $V_A + V_B$, in liters?

$25$ liters $30$ liters $40$ liters $50$ liters

51

A sphere with radius $R$ is melted down and recast into $8$ smaller, identical spheres. What is the ratio of the total surface area of the $8$ smaller spheres to the surface area of the original large sphere?

$2:1$ $4:1$ $8:1$ $\sqrt{2}:1$

52

A large cylindrical tank has two input valves, A and B. Valve A alone can fill the tank in $3$ hours. Valve B alone can fill the tank in $4$ hours. A drain pipe D can empty the tank in $x$ hours. If valves A and B are opened simultaneously, and the drain D is also open, the tank fills up in $5$ hours. If the tank is currently $75\%$ full, how long will it take the drain pipe D alone to empty the tank completely?

$2\frac{1}{2}$ hours $1\frac{1}{2}$ hours $2$ hours $1\frac{22}{23}$ hours

53

A train leaves Station P heading East at $60 \text{ km/h}$. Three hours later, a second, faster train leaves Station P heading East at $90 \text{ km/h}$. Station Q is $500 \text{ km}$ East of Station P. How far from Station Q will the faster train catch the slower train?

$40 \text{ km}$ $10 \text{ km}$ $50 \text{ km}$ $30 \text{ km}$

54

A warehouse contains $300$ boxes. Every box contains either electronics (E), clothes (C), or books (B), or some combination thereof. A survey revealed the following counts: $160$ contain E; $150$ contain C; $130$ contain B. $70$ contain E and C; $60$ contain C and B; $50$ contain E and B. If $30$ boxes contain all three items (E, C, and B), how many boxes contain exactly two types of items?

$120$ $90$ $150$ $100$

55

The product of three consecutive positive integers is $12$ times the middle integer. What is the sum of these three integers?

$12$ $9$ $6$ No such integer

56

In a coding system, a $3$-digit number $N$ must satisfy two conditions: it must be odd, and the sum of its digits must be $11$. The digits must be non-zero. How many such distinct $3$-digit numbers exist?

$18$ $25$ $14$ $16$

57

An arithmetic sequence represents the number of seats in a theater row by row. Row $1$ has $a$ seats, Row $2$ has $a+d$ seats, and so on. The theater has $20$ rows in total. If the ratio of the number of seats in Row $12$ to the number of seats in Row $4$ is $3:1$, what is the ratio of the total number of seats in the first $10$ rows to the total number of seats in the entire $20$ rows?

$11:42$ $11:80$ $1:4$ $7:30$

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