JAMB UTME - Mathematics - 2023

PQRS is a cyclic quadrilateral. Find x $\mathrm{<br>}$ + y

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∠PQR + ∠PSR = 180o (opp. angles of cyclic quad. are supplementary)

⇒ 5x $\mathrm{<br>}$ - y $\mathrm{<br>}$ + 10 + (-2x $\mathrm{<br>}$ + 3y $\mathrm{<br>}$ + 145) = 180

⇒ 5x $\mathrm{<br>}$ - y $\mathrm{<br>}$ + 10 - 2x $\mathrm{<br>}$ + 3y $\mathrm{<br>}$ + 145 = 180

⇒ 3x $\mathrm{<br>}$ + 2y $\mathrm{<br>}$ + 155 = 180

⇒ 3x $\mathrm{<br>}$ + 2y $\mathrm{<br>}$ = 180 - 155

⇒ 3x $\mathrm{<br>}$ + 2y $\mathrm{<br>}$ = 25 ----- (i)

∠QPS + ∠QRS = 180o (opp. angles of cyclic quad. are supplementary)

⇒ -4x $\mathrm{<br>}$ - 7y $\mathrm{<br>}$ + 150 + (2x $\mathrm{<br>}$ + 8y $\mathrm{<br>}$ + 105) = 180

⇒ -4x $\mathrm{<br>}$ - 7y $\mathrm{<br>}$ + 75 + 2x�
 + 8y $\mathrm{<br>}$ + 180 = 180

⇒ -2x $\mathrm{<br>}$ + y $\mathrm{<br>}$ + 255 = 180

⇒ -2x $\mathrm{<br>}$ + y = 180 - 255

⇒ -2x $\mathrm{<br>}$ + y $\mathrm{<br>}$ = -75 ------- (ii)

⇒ y $\mathrm{<br>}$ = -75 + 2x $\mathrm{<br>}$ -------- (iii)

Substitute (-75 + 2x $\mathrm{<br>}$) for y $\mathrm{<br>}$ in equation (i)

⇒ 3x $\mathrm{<br>}$ + 2(-75 + 2x $\mathrm{<br>}$) = 25

⇒ 3x $\mathrm{<br>}$ - 150 + 4x $\mathrm{<br>}$ = 25

⇒ 7x
 = 25 + 150

⇒ 7x $\mathrm{<br>}$ = 175

⇒ x=1757=25 $\mathrm{<br>}=\frac{175}{7}=25$

From equation (iii)

⇒ y $\mathrm{<br>}$ = -75 + 2(25) = -75 + 50

⇒ y $\mathrm{<br>}$ = -25

∴ x $\mathrm{<br>}$ + y $\mathrm{<br>}$ = 25 + (-25) = 0

The graph above depicts the performance ratings of two sports teams A and B in five different seasons

In the last five seasons, what was the difference in the average performance ratings between Team B and Team A?

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Average performance rating of Team B = 7+9+1+9+65=325 $\frac{7+9+1+9+6}{5}=\frac{32}{5}$ = 6.4

Average performance rating of Team A = 5+3+6+10+25=265 $\frac{5+3+6+10+2}{5}=\frac{26}{5}$ = 5.2

∴ The difference in the average performance ratings between Team B and Team A = 6.4 - 5.2 = 1.2

A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increase by 68 cm2 ${}^{\mathrm{<br>}}$ if we add 2 cm to each side of the rectangle. Find the length of the shorter side.

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Calculate the area of the composite figure above.

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200 tickets were sold for a show. VIP tickets costs ₦1,200 and ₦700 for regular. Total amount realised from the sale of the tickets was ₦180,000. Find the number of VIP tickets sold and the the number of regular ticket sold.

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To find the number of VIP tickets sold and the number of regular tickets sold, we can set up a system of equations based on the information given.
Let's assume that the number of VIP tickets sold is represented by 'V' and the number of regular tickets sold is represented by 'R'.
From the information given, we know that a total of 200 tickets were sold. Therefore, we can write the equation:
V + R = 200 -------------- (Equation 1)
We also know that VIP tickets cost ₦1,200 and regular tickets cost ₦700. The total amount realized from the sale of the tickets was ₦180,000. So, we can write another equation based on the ticket prices and total amount:
1200V + 700R = 180,000 -------------- (Equation 2)
Solving these two equations will give us the values of V (VIP tickets) and R (regular tickets).
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:
1. Solve Equation 1 for V in terms of R: V = 200 - R
2. Substitute this value of V in Equation 2: 1200(200 - R) + 700R = 180,000
Simplify the equation: 240,000 - 1200R + 700R = 180,000
Combine like terms: 500R = 60,000
Divide both sides by 500: R = 120
Now, we know the number of regular tickets sold is 120. We can substitute this value back into Equation 1 to find the number of VIP tickets:
V + 120 = 200
Subtract 120 from both sides: V = 200 - 120 V = 80
Therefore, the number of VIP tickets sold is 80 and the number of regular tickets sold is 120.
The correct answer is: VIP = 80, Regular = 120

Calculate, correct to three significant figures, the length AB in the diagram above.

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The area of a trapezium is 200 cm2 ${}^2$. Its parallel sides are in the ratio 2 : 3 and the perpendicular distance between them is 16 cm. Find the length of each of the parallel sides.

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To find the lengths of the parallel sides of the trapezium, we need to use the formula for the area of a trapezium.
The formula for the area of a trapezium is:
Area = (1/2) * (sum of parallel sides) * (perpendicular distance between them).
In this case, we are given that the area of the trapezium is 200 cm² and the perpendicular distance between the parallel sides is 16 cm.
Let's assume the lengths of the parallel sides are 2x and 3x.
Using the formula, we can write the equation as:
200 = (1/2) * (2x + 3x) * 16
Simplifying the equation, we have:
200 = (5x) * 16
Dividing both sides by 5, we get:
40 = 4x
Dividing both sides by 4, we get:
10 = x
So, the length of one of the parallel sides is 2x = 2 * 10 = 20 cm.
And the length of the other parallel side is 3x = 3 * 10 = 30 cm.
Therefore, the correct answer is:
10 cm and 15 cm.

The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6o, then the value of "n" is

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The angle of elevation and depression of the top and bottom of another building, measured from the top of a 24 m tall building, is 30° and 60°, respectively. Determine the second building's height.

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Two dice are tossed. What is the probability that the total score is a prime number.

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Total possible outcome = 6 x 6 = 36

Required outcome = 15

∴ Pr(E) = 1536=512

Solve the logarithmic equation: log2(6−x)=3−log2x

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To solve the logarithmic equation log2(6-x) = 3 - log2x, we can use the properties of logarithms.
First, let's simplify the equation by combining the logarithms on the right side:
log2(6-x) + log2x = 3
Next, we can use the logarithmic product rule, which states that logb(M * N) = logb(M) + logb(N), to combine the logarithms on the left side:
log2[(6-x) * x] = 3
To solve for x, we can rewrite the equation using exponential form. Since the base of the logarithm is 2, we can rewrite the equation as:
2^3 = (6-x) * x
Simplifying the left side gives us:
8 = (6-x) * x
Now, we have a quadratic equation. Let's expand the right side:
8 = 6x - x^2
Re-arranging the equation gives us:
x^2 - 6x + 8 = 0
To solve this quadratic equation, we can factor or use the quadratic formula. Factoring the equation gives us:
(x - 4)(x - 2) = 0
Setting each factor equal to zero gives us two possible solutions:
x - 4 = 0 or x - 2 = 0
Solving these equations gives us:
x = 4 or x = 2
Therefore, the solutions to the logarithmic equation log2(6-x) = 3 - log2x are:
x = 4 or x = 2

The third term of an A.P is 6 and the fifth term is 12. Find the sum of its first twelve terms

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A student is using a graduated cylinder to measure the volume of water and reports a reading of 18 mL. The teacher reports the value as 18.4 mL. What is the student's percent error?

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The student's percent error can be calculated by using the formula:
Percent Error = |(Measured Value - True Value) / True Value| * 100%
In this case, the measured value is 18 mL, and the true value reported by the teacher is 18.4 mL.
Let's substitute the values into the formula:
|(18 mL - 18.4 mL) / 18.4 mL| * 100%
Simplifying the formula:
|(-0.4 mL) / 18.4 mL| * 100%
Taking the absolute value of the fraction:
|-0.0217| * 100%
Calculating the absolute value:
0.0217 * 100%
Multiplying by 100%:
2.17%
Therefore, the student's percent error is 2.17%.
Note: It is important to note that in this case, the student's measurement was lower than the true value, causing a negative difference. By taking the absolute value of the fraction in the formula, we ignore the negative sign and only consider the magnitude of the error.

Which inequality describes the graph above?

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The inequality that describes the graph above is 5y + 4x <= 20.

Let's break down the reasoning behind this answer:

In the given inequality options, we can see that the coefficients in front of x and y are 5 and 4 respectively. In the graph, we can observe that the line is steeper in the y-direction compared to the x-direction. This means that the slope of the line is greater in the y-direction compared to the x-direction.

The inequality sign for the equation is less than or equal to in the selected option of 5y + 4x <= 20. This suggests that the shaded region below or on the line represents the solution. This is because any point below or on the line will satisfy the condition.

Therefore, the graph above corresponds to the inequality 5y + 4x <= 20.

Find the volume of a cone which has a base radius of 5 cm and slant height of 13 cm.

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A ship sets sail from port A (86o ${}^{\mathrm{<br>}}$N, 56o ${}^{\mathrm{<br>}}$W) for port B (86o ${}^{\mathrm{<br>}}$N, 64o ${}^{\mathrm{<br>}}$W), which is close by. Find the distance the ship covered from port A to port B, correct to the nearest km.

[Take π = 3.142 and R = 6370 km]

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Find the volume of the cylinder above
[Take π=22/7]

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The second term of a geometric series is −2/3 and its sum to infinity is 3/2. Find its common ratio.

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Calculate, correct to three significant figures, the length of the arc AB in the diagram above.
[Take π=22/7]

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A man sells different brands of an items. 1/9 ${}^1{/}_9$ of the items he has in his shop are from Brand A, 5/8 ${}^5{/}_8$ of the remainder are from Brand B and the rest are from Brand C. If the total number of Brand C items in the man's shop is 81, how many more Brand B items than Brand C does the shop has?

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Study the given histogram above and answer the question that follows.

What is the total number of students that scored at most 50 marks?

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Total number of students that scored at most 50 marks = 100 + 80 + 60 + 40 + 80 = 360

A student pilot was required to fly to an airport and then return as part of his flight training. The average speed to the airport was 120 km/h, and the average speed returning was 150 km/h. If the total flight time was 3 hours, calculate the distance between the two airports.

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Give the number of significant figures of the population of a town which has approximately 5,020,700 people

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The two trailing zeros in the number are not significant, but the other five are, making it a five-figure number.

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To find the range of possible values for the area of the rectangular plot of land, we need to consider the possible range of lengths of the sides.
Given that the lengths of the sides are 38 m and 52 m, correct to the nearest meter, there is a possibility of some error in the measurements. So, let's consider the minimum and maximum possible lengths for each side.
For the first side with a length of 38 m, the minimum possible length would be 37.5 m (rounded down) and the maximum possible length would be 38.5 m (rounded up).
For the second side with a length of 52 m, the minimum possible length would be 51.5 m (rounded down) and the maximum possible length would be 52.5 m (rounded up).
Now, let's calculate the range of possible areas using these minimum and maximum lengths.
The minimum area of the rectangle would be (37.5 m) * (51.5 m) = 1931.25 m² (rounded to the nearest 0.01 m²).
The maximum area of the rectangle would be (38.5 m) * (52.5 m) = 2021.25 m² (rounded to the nearest 0.01 m²).
Therefore, the range of possible values for the area of the rectangle is 1931.25 m² ≤ A < 2021.25 m².
So, the correct answer is 1931.25 m² ≤ A < 2021.25 m².

Use the graph of sin (θ) above to estimate the value of θ when sin (θ) = -0.6 for 0o≤θ≤360o

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A circle has a radius of 13 cm with a chord 12 cm away from the centre of the circle. Calculate the length of the chord.

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To calculate the length of the chord in a circle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, we can consider the radius of the circle as one side of the right triangle, the distance from the center of the circle to the chord as the other side, and the chord itself as the hypotenuse.

Let's denote the radius as r, the distance from the center to the chord as d, and the length of the chord as c.

Using the Pythagorean theorem, we have:

r² = d² + (c/2)²

Since we know the value of the radius (13 cm) and the distance from the center to the chord (12 cm), we can substitute those values into the equation:

13² = 12² + (c/2)²

Simplifying the equation, we get:

169 = 144 + (c/2)²

Subtracting 144 from both sides, we have:

25 = (c/2)²

Taking the square root of both sides, we get:

5 = c/2

Multiplying both sides by 2, we find:

c = 10 cm

Therefore, the length of the chord in the circle is 10 cm.

Find the area, to the nearest cm2 ${}^{\mathrm{}}$, of the triangle whose sides are in the ratio 2 : 3 : 4 and whose perimeter is 180 cm.

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Find the area and perimeter of a square whose length of diagonals is 202–√ cm.

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Tickets for the school play were priced at ₦520.00 each for adults and ₦250.00 each for kids. How many kids' tickets were sold if the total sales were ₦171,000.00 and there were 5 times as many adult tickets sold as children's tickets?

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To find the number of kids' tickets sold for the school play, we'll need to break down the given information:
1. The price of an adult ticket is ₦520.00. 2. The price of a kids' ticket is ₦250.00. 3. The total sales from all the tickets sold is ₦171,000.00. 4. The number of adult tickets sold is 5 times the number of kids' tickets sold.
Let's use a logical approach to solve this problem step by step.
Step 1: Let's assume that "x" represents the number of kids' tickets sold.
Step 2: To find the number of adult tickets sold, we'll need to multiply the number of kids' tickets sold by 5, since there were 5 times as many adult tickets sold as children's tickets. So the number of adult tickets sold would be 5x.
Step 3: Now let's calculate the total sales from the tickets sold. To do this, we'll need to multiply the price of an adult ticket (₦520.00) by the number of adult tickets sold (5x), and the price of a kids' ticket (₦250.00) by the number of kids' tickets sold (x). Then we can add these two values together to get the total sales.
Total Sales = (₦520.00 * 5x) + (₦250.00 * x) = ₦2600.00x + ₦250.00x = ₦2850.00x
Step 4: We know that the total sales is ₦171,000.00. So we can set up an equation using the above expression for total sales:
₦171,000.00 = ₦2850.00x
Step 5: To isolate "x" (the number of kids' tickets sold), we need to divide both sides of the equation by ₦2850.00:
₦171,000.00 / ₦2850.00 = x
Step 6: Simplifying the equation, we find:
x = 60
Therefore, the number of kids' tickets sold for the school play is 60.

The perimeter of an isosceles right-angled triangle is 2 meters. Find the length of its longer side.

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Evaluate 

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Find the value of x in the diagram above

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Intersecting Chords Theorem states that If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal.

⇒ AE * EB = CE * ED

⇒ 6 * x $\mathrm{<br>}$ = 4 * (x $\mathrm{<br>}$ + 5)

⇒ 6x $\mathrm{<br>}$ = 4x $\mathrm{<br>}$ + 20

⇒ 6x $\mathrm{<br>}$ - 4x $\mathrm{<br>}$ = 20

⇒ 2x $\mathrm{<br>}$ = 20

∴ x=202 $\mathrm{<br>}=\frac{20}{2}$ = 10 units

Find the volume of the composite solid above.

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Volume of the composite solid = Volume of A + Volume of B

Volume of a cuboid = length x breadth x height

Volume of A = 6 x 26 x 8 = 1248 cm3 ${}^3$

Volume of B = 6 x 10 x 22 = 1320 cm3 ${}^3$

∴ Volume of the composite solid = 1248 + 1320 = 2568 cm3

Bello buys an old bicycle for ₦9,200.00 and spends ₦1,500.00 on its repairs. If he sells the bicycle for ₦13,400.00, his gain percent is

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To calculate the gain percent, we need to compare the profit made to the cost price.
The cost price is the amount Bello spent on buying the bicycle and repairing it, which is ₦9,200.00 + ₦1,500.00 = ₦10,700.00.
The profit made is the amount Bello sold the bicycle for minus the cost price, which is ₦13,400.00 - ₦10,700.00 = ₦2,700.00.
To find the gain percent, we need to divide the profit made by the cost price, and then multiply by 100.
So, gain percent = (profit / cost price) * 100 = (₦2,700.00 / ₦10,700.00) * 100
Calculating this gives us gain percent = 25.2336448598
Rounded to two decimal places, the gain percent is approximately 25.23%.
Therefore, the correct answer is 25.23%.

The line 3y + 6x = 48 passes through the points A(-2, k) and B(4, 8). Find the value of k.

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To find the value of k, we can use the given information that the line passes through points A(-2, k) and B(4, 8).
The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.
First, let's find the slope (m) of the line using the coordinates of the two points. The formula for slope is (y2 - y1) / (x2 - x1).
Substituting the coordinates of point A and B into the formula, we have: m = (8 - k) / (4 - (-2)) m = (8 - k) / 6
The line also passes through point A(-2, k), so we can substitute these values into the slope-intercept form equation: k = m(-2) + b
Now we have two equations: k = m(-2) + b m = (8 - k) / 6
To simplify the situation, we need to eliminate the variable b by isolating it in the first equation. Let's solve the first equation for b: b = k + 2m
Now we have: m = (8 - k) / 6 b = k + 2m
Next, substitute the expression for b into the second equation: m = (8 - k) / 6 k + 2m = (8 - k) / 6
To solve this equation for k, we will multiply both sides by 6 to eliminate the denominator: 6k + 12m = 8 - k
To simplify the equation, we bring like terms together: 6k + k = 8 - 12m 7k = 8 - 12m 7k + 12m = 8
Now, we have a linear equation in two variables (k and m). To solve for k, we need to know the value of m.
Assuming we know the value of m, we can substitute it into the equation 7k + 12m = 8 and solve for k.
Based on the given options, we can assume a value for k and calculate the corresponding value of m using the equation m = (8 - k) / 6.
Let's try k = 20: m = (8 - 20) / 6 m = -12/6 m = -2
Now we substitute m = -2 into the equation 7k + 12m = 8: 7k + 12(-2) = 8 7k - 24 = 8 7k = 32 k = 32/7
Therefore, when k = 20, the equation satisfies the given line equation and passes through points A(-2,20) and B(4,8).
Hence, the value of k is 20 when the line passes through points A(-2, k) and B(4, 8).

Express 16.54 x 10−5 - 6.76 x 10−8 + 0.23 x 10−6 in standard form

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Find the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually.

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To find the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually, we can use the formula for compound interest:
CI = P(1 + r/n)^(nt) - P
Where: - CI is the compound interest - P is the principal amount (₦15,700 in this case) - r is the annual interest rate (8% or 0.08 as a decimal) - n is the number of times the interest is compounded per year (since it's compounded annually, n is 1) - t is the number of years (2 in this case)
Now we can substitute the values into the formula:
CI = ₦15,700(1 + 0.08/1)^(1*2) - ₦15,700
Simplifying the equation:
CI = ₦15,700(1.08)^2 - ₦15,700
CI = ₦15,700(1.1664) - ₦15,700
CI = ₦18,312.48 - ₦15,700
CI = ₦2,612.48
Therefore, the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually is ₦2,612.48.

An article when sold for ₦230.00 makes a 15% profit. Find the profit or loss % if it was sold for ₦180.00

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To find the profit or loss percentage, we need to compare the selling price to the cost price. Let's call the cost price of the article C.

We know that when the article is sold for ₦230.00, it makes a 15% profit. Profit is calculated by subtracting the cost price from the selling price. So, we can write the equation:

Selling Price - Cost Price = Profit

₦230.00 - C = 15% of C

Since we want to find the profit or loss percentage when the article is sold for ₦180.00, we can use the same equation:

₦180.00 - C = x% of C

where x is the profit or loss percentage we want to find.

Now, let's solve these equations to find the value of C and x.

From the first equation:

₦230.00 - C = 0.15C

₦230.00 = 0.15C + C

₦230.00 = 1.15C

To solve for C:

C = ₦230.00 / 1.15

C = ₦200.00

Now, let's substitute the value of C into the second equation:

₦180.00 - ₦200.00 = x% of ₦200.00

-₦20.00 = x% of ₦200.00

To solve for x:

x% = (-₦20.00 / ₦200.00) * 100

x% = -10%

Therefore, when the article is sold for ₦180.00, it results in a 10% loss.

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To solve this problem, let's first calculate the total work that needs to be done. We can do this by multiplying the number of men, the number of hours they work per day, and the number of days it takes them to complete the work.
For the first scenario, we have: - 12 men working together for 8 hours a day - It takes them 4 days to finish the work
So, the total work done by these 12 men can be calculated as: work = (12 men) * (8 hours/day) * (4 days) = 384 man-hours
Now, let's find out how long it would take 4 men working 16 hours a day to complete the same piece of work.
If 12 men can do the work in 4 days, it means that the total man-hours required to complete the work is the same for both scenarios.
Let's use the same formula to calculate the total work for the second scenario: work = (4 men) * (16 hours/day) * (x days) = 384 man-hours
We need to solve for x, which represents the number of days required by 4 men to complete the work.
Dividing both sides of the equation by (4 men) and (16 hours/day), we get: x = 384 man-hours / (4 men * 16 hours/day) x = 384 / 64 x = 6
So, it will take 4 men working 16 hours a day approximately 6 days to complete the same piece of work.
Therefore, the correct answer is 6 days.

Find the value of the angle marked x in the diagram above

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