Unit 2 - Practice Quiz

To convert a percentage to a fraction, you place the number over 100. So, . Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, 25, gives .

To find the percentage of a number, you can convert the percentage to a decimal and multiply. . So, .

A ratio shows the relative sizes of two or more values. It is fundamentally a comparison by division, often written as or .

To simplify a ratio, find the greatest common divisor (GCD) of the numbers. The GCD of 16 and 24 is 8. Divide both parts of the ratio by 8: and . The simplified ratio is 2:3.

In a direct proportion, if one quantity increases, the other quantity increases at the same rate (and vice versa). The relationship is . If becomes , then the new will be , so is also doubled.

In a proportion , the outer terms ( and ) are called the 'extremes', and the inner terms ( and ) are called the 'means'.

Direct variation between two variables and is expressed by the equation , where is the non-zero constant of variation.

This is a classic example of inverse variation. As one quantity (number of workers) increases, the other quantity (time taken) decreases, assuming the total amount of work remains constant.

An Arithmetic Progression is a sequence of numbers such that the difference between the consecutive terms is constant. This constant is called the common difference ().

The common difference () in an AP is found by subtracting any term from its succeeding term. For example, or .

First, find the common difference: . To find the next term, add the common difference to the last known term: .

A Geometric Progression is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ().

The common ratio () in a GP is found by dividing any term by its preceding term. For example, or .

First, find the common ratio: . To find the next term, multiply the last known term by the common ratio: .

The discount amount is 15% of $80. This is calculated as .

Since the value for 'b' is the same (4) in both ratios, we can directly combine them. The continuous ratio is . Therefore, the ratio is 3:5.

This is a direct proportion. First, find the cost of one pen: per pen. Then, multiply by the desired number of pens: .

For inverse variation, . First, find the constant : . Now, use this constant to find the new value of when : , so .

The formula for the term () of an AP is , where is the first term, is the term number, and is the common difference. The other options are for the sum of an AP or terms/sum of a GP.

In the formula for the term of a GP, , the variable '' stands for the first term of the sequence.

Let the original salary be .
After a 20% increase, the new salary becomes .
This new salary is then decreased by 25%. The final salary is .
The net change is the difference between the final salary and the original salary: .
This represents a 10% decrease from the original salary.
Alternatively, using the formula for successive percentage changes : . A negative sign indicates a decrease.

Let the total number of votes be .
The losing candidate got 42% of the votes, so the winning candidate got of the votes.
The margin of loss (or victory) is the difference in the percentage of votes between the two candidates.
Margin = .
We are given that this margin is equal to 1,408 votes.
So, 16% of is 1,408.
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Thus, the total number of votes polled was 8,800.

Let the monthly incomes be and .
Let their monthly expenditures be and .
Savings = Income - Expenditure.
For the first person: (Equation 1)
For the second person: (Equation 2)
We need to solve this system of linear equations. Multiply Eq. 1 by 9 and Eq. 2 by 7:


Subtracting the second new equation from the first:

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Their incomes are and .
Their combined monthly income is 9,000.

Let the number of 5x$, $6x8x$ respectively.
The total value of the coins is the sum of the values of each type of coin.
Value = .
Total Value = .
We are given that the total amount is $600.
So, , which implies .
The number of 50 paise coins is .
Number of 50 paise coins = .

Given the proportion .

1. Componendo: Adding 1 to both sides gives , which simplifies to .
2. Dividendo: Subtracting 1 from both sides gives , which simplifies to .
3. Componendo and Dividendo: Dividing the result of Componendo by the result of Dividendo:
   
   This simplifies to . This is the correct property.

Let be the cost and be the weight. The relationship is , where is the constant of proportionality.
First, find using the given information: .
So, .
The cost formula is .
The original value of the 10-gram diamond was $32,000.
After breaking, we have two pieces:
Cost of the 4-gram piece = 5,120$.
Cost of the 6-gram piece = 11,520$.
The total value of the broken pieces is 16,640$.
The loss in value is the original value minus the new total value.
Loss = 16,640 = $15,360.

This is a problem of inverse variation (more men, less time).
The total food units available can be thought of as (Number of soldiers) (Number of days).
Total food units = man-days.
For the first 3 days, food consumed = man-days.
Remaining food units = man-days.
After 3 days, the number of soldiers becomes .
Now, we need to find how many days () the remaining food will last for 800 soldiers.
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The remaining food will last for 15 more days.

The relationship is given by the formula , where is the constant of variation.
First, use the initial conditions to find .
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So the specific formula is .
Now, use the new conditions to find the new pressure.
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The new pressure will be 80.

Let the first term be and the common difference be . The nth term is .
Given :
(Equation 1).
Given :
(Equation 2).
Subtracting Eq. 1 from Eq. 2:
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Substitute into Eq. 1:
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The sum of the first n terms is .
We need to find :
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The nth term () of a series can be found by the formula .
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This expression for is a linear function of , which is a characteristic property of an AP. The common difference is the coefficient of , which is 6. So, the series is an AP.
To find the 15th term, substitute into the formula for .
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Let the first term be and the common ratio be . The first 5 terms of the GP are:
, , , , .
We are given that the 3rd term, .
The product of the first 5 terms is:
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This can be rewritten as .
Since we know , the product is .
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Thus, the product of the first 5 terms is 1024.

The total distance is the sum of the distances travelled downwards and upwards.
Initial downward distance = 120 m.
After the first bounce, it travels up and then down.
Upward distance sequence: , , , ...
Downward distance sequence (after initial drop): , , , ...
This forms an infinite geometric series with first term and common ratio .
The sum of an infinite GP is , where .
The sum of all upward distances is m.
The sum of all downward distances (after the first drop) is the same, m.
Total distance = Initial drop + Total upward + Total downward = meters.
Alternatively, using a formula: Total Distance = meters.

The student scored 178 marks and failed by 22 marks.
This means the passing marks are .
We are told that the passing marks are 40% of the maximum marks.
Let the maximum marks be .
Then, 40% of .
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Therefore, the maximum marks in the examination are 500.

To find the ratio , we can multiply the given ratios.



To find , we multiply these fractions:


We can simplify before multiplying. The 6 in the numerator and 3 in the denominator can be simplified.
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So, the ratio is 16:35.

Let be the time taken and be the number of workers.
The relationship is inverse variation, so or , where is a constant representing the total work in worker-days.
First, find the constant using the initial data.
worker-days.
Now, use this constant to find the time it takes for 12 workers.
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days.
It will take 12 workers 10 days to complete the task.

The savings form an Arithmetic Progression (AP).
The first term is .
The common difference is .
The sum of savings is .
We need to find the number of months, .
The formula for the sum of an AP is .
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Divide the entire equation by 50:
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Rearrange into a quadratic equation: .
We need to factor this equation. We look for two numbers that multiply to -540 and add to 7. These numbers are 27 and -20.
So, .
The possible values for are -27 and 20. Since the number of months cannot be negative, .
His total savings will be $13,500 in 20 months.

Let the first term be and the common ratio be .
Sum of first two terms: (Equation 1).
Fifth term is 4 times the third term: .
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Since and , we can divide by :
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Since the common ratio is positive, .
Now substitute into Equation 1:
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The first term is -4/3.

Let the two original numbers be and .
When 9 is subtracted from each, the new numbers are and .
The new ratio is given as 12:23.
So, .
Cross-multiply to solve for :
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The original numbers were and .
The smaller number is .

The sum of the interior angles of a polygon with sides is given by .
For a pentagon, , so the sum of angles is .
The angles are in an AP. Let the smallest angle be . We are given .
The 5 angles are .
The sum of these angles is .
Using the sum formula :
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The largest angle is the 5th term, which is .
Largest Angle = .

This is an infinite geometric series.
The first term is .
To find the common ratio , divide the second term by the first term:
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The sum of an infinite geometric series exists if . Here, , so the sum exists.
The formula for the sum is .
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Let the initial revenue in 2020 be .
After a increase, the revenue in 2021 is .
After a decrease from the 2021 level, the revenue in 2022 is .
Given that the revenue in 2022 is the same as in 2020, .
So, .
Assuming , we can divide by : .
Expanding the left side: .
Subtracting 1 from both sides: .
Multiply the entire equation by 10000 to eliminate fractions: .
Now, we need to solve for .
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Let the quantities taken from vessels A and B be and liters respectively.
In liters from A: Spirit = , Water = .
In liters from B: Spirit = , Water = .
Total Spirit in the first new mixture = .
Total Water in the first new mixture = .
Now, 11 liters of pure spirit are added.
Final quantity of Spirit = .
Final quantity of Water = .
The final ratio of Spirit to Water is 4:1.
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and . Ratio is 391:246. , . Not simplifiable. Let's re-calculate sums.
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From A(21L): S=15, W=6.
From B(28L): S=14, W=12.
Initial mix: S=15+14=29, W=6+12=18. Total=47L.
Add 11L spirit. Final S = 29+11=40. Final W=18. Ratio is 40:18 = 20:9. Not 4:1.
Okay, let me debug the equation: . . . This is mathematically correct. The numbers in the question must be flawed. Let's re-engineer.
We want to be an integer. Let's say we want . Then . So we'd need 1001 to be 4151. Let's try to make the coefficients integers. Let and be and . No, that's too complex.
Spirit = .
Water = .
Now add 11L spirit: .
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. . . Sum=391. Correct. . . Sum=246. Correct. . Correct. . Correct. The numbers are just difficult. Let's re-create the question with cleaner numbers.
New setup: A (3:1 Spirit:Water), B (5:3 S:W). Mix in ratio 2:3. Add 7L of Water. Final ratio is 1:1. Find quantity from A.
Let quantities be .
From A: S=, W=.
From B: S=, W=.
Total S = .
Total W = .
Add 7L water: Final S=, Final W=.
Ratio is 1:1 .
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Let's make the final ratio 20:9. Then the answer is 21. That's a valid hard question. Let's change the question text. "If the final mixture has spirit and water in the ratio 20:9". No, let me find numbers that work for 4:1.
We need . So . This is fixed. Let's change the added spirit. We need to be an integer multiple of . Let's change the amount of spirit added, say . Then . So should be a multiple of 593. That's too large. Let's change the final ratio. Let's make it . Then . . Let . Then . Still not a nice ratio.
My conclusion is the initial problem design had a calculation error. I will replace it with a corrected version that works.
Let's try: A(5:1), B(3:5). Mix ratio 2:3. Add 3L water. Final ratio 1:1. Find Q from A.
Quantities: .
From A(2x): S=, W=.
From B(3x): S=, W=.
Total S = .
Total W = .
Add 3L water: Final S=, Final W=.
Ratio 1:1 . Still not integer.
Okay, final attempt to fix the original question. I'll re-calculate everything. Ratio from A and B is 3:4. Let . Then . S_A=15, W_A=6. S_B=14, W_B=12. Total S=29, W=18. Let's say we add spirit and get 4:1 ratio. . So if we change 11 liters to 43 liters, the answer is 21. Let's stick with that. Question will be

Let each of the given ratios be equal to a constant .
1)
2)
3)
Adding these three equations together:


Since it is given that , we can divide both sides by to get .
Now substitute back into the original relations:
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2)
3)
Now we can evaluate the required expression by substituting these values:


Assuming are non-zero, the expression simplifies to 8.

Let be the cost and be the weight of the diamond. According to the problem, , so for some constant .
The original diamond weighs 8 grams. Its original cost is .
Let the diamond break into two pieces with weights and . We know that .
The cost of the two pieces together is .
The loss in value is .
The percentage loss is given as 37.5%, which is equal to .
So, .
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Substitute into this equation:
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Let the first term of the AP be and the common difference be . The sum of the first terms is given by .
We are given:
1)
2)
Subtracting the second equation from the first:




Since , we can divide the entire equation by :

Now, we need to find the sum of the first terms, which is .
From the equation we just derived, we know that the term is equal to -2.
Therefore, .

Let the infinite geometric series be where .
The sum of this series is .
We are given , so (Equation 1).
The series of the squares of the terms is , which is .
This is also a geometric series with the first term and the common ratio . Since , we have .
The sum of this new series is .
We are given , so (Equation 2).
We can write Equation 2 as . This can be rewritten as .
From Equation 1, we know . Substituting this in:
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(Equation 3).
Now we have a system of two linear equations in terms of and :
From (1):
From (3):
Equating the expressions for : .
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Now, substitute the value of back into either equation to find . Using :
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So, the first term of the series is 5.

Let the AP have first term and common difference . The sum formula is .
We are given .
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As , we can cancel from both sides:
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Rearrange to relate and :
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Now, we need to find the ratio .
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Cancel : .
Now substitute into this expression:
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Since , we can cancel :
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Let the work rate of one man be units per day. The total work planned is proportional to Men Days Efficiency.
In the first 30 days, 100 men worked. Let their efficiency be .
Work done, man-days.
This represents half of the total work. So, Total Work man-days.
Remaining work, man-days.
Remaining time = Total time - Time elapsed = days.
The new efficiency of each man is .
Let the new total number of men be . We need to complete the remaining work in the remaining time with the new efficiency.
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men.
The new total number of men required is 240.
The number of additional men to employ is .

We are given three conditions:

1. are in AP .
2. are in GP .
3. are in HP . (The middle term is the harmonic mean of the other two).
   We want to find a relationship between . We can do this by eliminating and .
   First, substitute the expression for from (1) into the expression for from (2):
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   Now we have two expressions for . Equating them:
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   Assuming , we can divide both sides by :
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   Cross-multiply:
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   Subtracting from both sides gives:
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   This is the condition for three numbers to be in Geometric Progression (GP). Therefore, are in GP.

Let the annual incomes of A and B be and respectively.
Let their annual expenditures be and respectively.
Savings = Income - Expenditure.
Savings of A, .
Savings of B, .
It is given that A saves one-third of his income.
So, .
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Multiply by 3 to clear the fraction:
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From this, we can establish a relationship between and . Let's express in terms of : .
Now we can find the savings of B in terms of :
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We already know the savings of A: .
Now find the ratio of their savings, .
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Thus, the ratio of their savings is 12:13.

The total distance is the sum of the distances traveled downwards and the distances traveled upwards.
Initial downward distance = 120 m.
After the first bounce, it goes up by and then comes down by the same amount.
After the second bounce, it goes up by and then comes down by the same amount.
And so on.
Let's separate the total distance into two parts:

1. The initial drop: m.
2. The sum of all subsequent upward and downward movements. This part forms an infinite geometric series.
   The distance traveled up after the first bounce is m. The distance traveled down is also 96 m.
   The distance traveled up after the second bounce is . The distance traveled down is also .
   The total distance of the bounces can be seen as .
   The upward movements form a GP:
   This is an infinite GP with first term and common ratio .
   The sum of this infinite GP is m.
   This is the total distance traveled upwards. The total distance traveled downwards (excluding the initial drop) is also 480 m.
   Total distance = Initial Drop + Total Upward + Total Downward (after first drop)
   Total distance = m.
   Alternatively, Total distance = m.

Let the total number of enrolled voters be .
Number of voters who did not vote = 15% of .
Number of voters who polled their votes (votes polled) = .
Number of invalid votes = 10% of votes polled = .
Number of valid votes = Votes polled - Invalid votes = .
Alternatively, valid votes are 90% of polled votes: .
The successful candidate got 55% of the valid votes.
The other candidate (loser) got of the valid votes.
The winner's majority is the difference in their vote percentages, applied to the valid votes.
Majority = (55% - 45%) of valid votes = 10% of valid votes.
We are given that the majority is 3060 votes.
So, .
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To simplify, notice . So, .
Thus, the total number of enrolled voters was 40,000.

This problem describes a linear relationship, not a simple direct proportion. Let the total expense be , the constant part be , the variable part per member be , and the number of members be . The relationship is given by the equation: .
We are given two scenarios:

1. For , .
2. For , .
   We have a system of two linear equations with two variables, and . We can solve it by subtracting the first equation from the second:
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   . This is the variable cost per member.
   Now substitute back into the first equation to find the constant cost :
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   . This is the fixed expense.
   The expense formula for this family is .
   We need to find the expense for a family of 5 members, so we set .
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   The total expense for a family of 5 members is $12,400.

The monthly savings form an Arithmetic Progression (AP) with:
First term, .
Common difference, .
We want to find the smallest integer (number of months) for which the sum of savings is greater than 15,000.
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The formula for the sum of an AP is .
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Divide by 10: .
To find when this inequality holds, let's find the positive root of the corresponding equation using the quadratic formula .
(we only need the positive root).
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We know that , so is slightly greater than 190. Let's approximate it as 190.5.
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Since the quadratic is an upward-opening parabola, the inequality will be true for greater than the positive root. Since must be an integer, the smallest integer value for is 29.
Let's verify:
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Therefore, the cumulative savings will first exceed $15,000 in the 29th month.

The relationship is given by the formula , where is a constant.
Let the initial pressure, temperature, and volume be . So, .
The temperature is increased by 20%. The new temperature is:
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The volume is decreased by 10%. The new volume is:
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The new pressure is given by:
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We can rearrange this to relate it to :
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Simplifying the fraction: .
So, .
The percentage change in pressure is calculated as .
Percentage change = .
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This represents a 33.33% increase in pressure.

Let the five terms of the GP be represented symmetrically around the third term, . Let the common ratio be .
The terms are: .
The product of these five terms is .
We are given that this product is 1024.
. We know that , so .
The third term of the GP is 4.
The first term is .
We are given that the third term is the square of the first term:
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Since , we can divide by :
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Substituting : . This means , so .
We need to find the 5th term of the GP, which is .
5th term = .

Initial total volume of mixture = 150 liters.
Initial quantity of milk = 60% of 150 = liters.
Initial quantity of water = liters.
'x' liters of milk are added. New quantity of milk = .
'x-10' liters of water are added. New quantity of water = .
The new total volume of the mixture is (Initial Volume) + (Milk Added) + (Water Added).
New Total Volume = .
The final concentration of milk is 70%.
This means: .
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Cross-multiply to solve for :
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Let's make final concentration 65%. . . . . Still doesn't work. The final concentration must be between the initial (60%) and the added part. The added part is x milk and x-10 water. This ratio is x/(2x-10), which for large x is ~50%. So the final concentration should go down. Let's make the final concentration 55%.
. . . . This works. Ok, let me re-engineer the original question to make the numbers clean. Let's start with the answer .
New milk = . New water = . New total = . Final concentration = . Let's set the final concentration to be . No, that's ugly.
How about we add x milk and x water? Then new milk=90+x, new water=60+x, new total=150+2x. Final conc = 70%. . . . No.
Initial: 120L, Milk:Water = 2:1. So Milk=80, Water=40. Add 'x' L milk and 'x' L water. Final milk conc becomes 60%.
'In a 120-liter mixture of milk and water, the ratio of milk to water is 2:1. 'x' liters of pure milk and 'x' liters of pure water are added. If the final concentration of milk becomes 60%, what is the value of 'x'?'

Let the Cost Price (CP) of the item be $100.
Marked Price (MP) is 60% above CP: MP = .
The first discount is 25% on the MP.
Price after first discount = .
Let the second discount be . This discount is applied to the price of $120$.
Final Selling Price (SP) = .
The shopkeeper makes a profit of 12% on the CP.
Final SP = CP + 12% of CP = .
Now, we equate the two expressions for the final SP:
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To simplify the fraction: .
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. Uh oh, calculation error again. Let's check. CP=100, MP=160. First discount 25%. Price = 160 0.75 = 120. Profit is 12%, so SP=112. Second discount d%. . . This is correct. My options are wrong. Let's re-engineer.
Let second discount be 10%. Then SP = . Profit = (108-100)/100 = 8%. So if the final profit is 8%, the discount is 10%. Let's change the profit in the question to 8%.
Original Question: '...profit of 12%...' Final SP = 112. Price after 1st disc = 120. Discount amount = 120 - 112 = 8. Discount % = . The question or options are definitely flawed. I will fix the question to yield one of the answers.
Let's change the profit to 10%. Final SP = 110. Price after 1st discount = 120. Discount amount = 10. Discount % = (10/120)100 = 8.33%.
Let's change the first discount to 20%. MP=160. Price after 1st discount = . Final profit 12%, SP=112. Discount amount = 128-112=16. Discount % = . This is a good number. Let's change the first discount to 20%.
Final try. Let's keep first discount at 25%. MP=160, Price=120. Let final profit be 8%. SP=108. Discount = 120-108 = 12. Discount % = (12/120)*100 = 10%. This works perfectly with the 10% option. So I will change the profit percentage in the question to 8%.

Let the number of sides of the polygon be . The number of interior angles is also .
The sum of the interior angles of a convex polygon with sides is given by the formula .
The angles are in an arithmetic progression (AP) with:
First term, .
Common difference, .
Number of terms = .
The sum of this AP is .
Equating the two formulas for the sum of the angles:
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Divide the equation by 5:
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This is a quadratic equation for . We can solve it by factoring:
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This gives two possible values for : or .
However, in a convex polygon, every interior angle must be less than 180 degrees. Let's check the largest angle for each possible value of .
The largest angle is the -th term of the AP, given by .
Case 1: .
Largest angle = . Since , this is a valid solution.
Case 2: .
Largest angle = . Since , a convex polygon cannot have such an interior angle. Therefore, is not a valid solution.
The only valid number of sides is 9.

Let the infinite geometric progression be , where is the first term and is the common ratio. For the sum to converge, we must have .
Let's consider any term in the series, say the -th term, .
The sum of all the terms that follow it is the sum of an infinite GP starting from the -th term.
The subsequent series is
This is an infinite GP with the first term and common ratio .
The sum of this subsequent series is .
The condition given is that each term is equal to three times the sum of all the terms that follow it.
So, .
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Since the first term is not zero, we can divide both sides by . Also, we can divide by (as long as ):
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Now, we solve for .
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This value satisfies the condition , so it is a valid common ratio.

Loss = 7248. Original Value = 10872. New Value = 10872 - 7248 = 3624.
. . Not a nice number.
Let's assume the loss calculation formula is wrong. Loss factor = . Loss = . Still 6644. The value $7,248 is exactly of $10,872\frac{22}{36} = 11/18
eq 2/3V{orig}=k(3w)^2=9kw^2$. $V{new}=k(w^2+w^2+w^2)=3kw^26kw^26/9=2/310872 \times 2/3 = 7248$. This makes a much better question.
So I'll rephrase the question: '...broke into three pieces of equal weight.'