Category: Civil Services Exam

Ratio and Proportion

Ratio and proportion are mathematical concepts that are closely related. A ratio is a comparison of two or more quantities, while a proportion is an equation that states two ratios are equal.

A ratio expresses the relationship between two or more numbers. It can be written in different forms, such as using the word “to” or a colon (:). For example, the ratio of girls to boys in a class could be 2:3 or 2 to 3.

Proportion, on the other hand, compares two ratios and states that they are equal. It can be expressed using the symbol “=”, such as in the equation 2:3 = 4:6. This means that the ratio of girls to boys is the same as the ratio of 4:6.

Ratios and proportions are used in various fields, such as cooking, finance, construction, and statistics. They help in making comparisons, solving problems, and making predictions.

To solve problems involving ratios and proportions, you can use cross-multiplication, where you multiply the numerator of one ratio with the denominator of the other ratio. This can help you find the unknown value or verify if two ratios are proportional.

Here are a few examples of ratios:

1. Recipe Ratios: In a recipe, ingredients are often listed in ratios to ensure the right proportions. For example, a pancake recipe might call for a ratio of 2 cups of flour to 1 cup of milk.
2. Financial Ratios: Financial ratios are used to analyze a company’s financial performance. For instance, the debt-to-equity ratio compares the company’s total debt to its shareholder’s equity.
3. Speed Ratios: Speed ratios compare the speeds of different objects or vehicles. For instance, if a car travels 60 miles in 2 hours, its speed ratio would be 60 miles to 2 hours, or 30 miles per hour.
4. Scale Ratios: Scale ratios are used in maps, models, and architectural blueprints to represent the relationship between the measurements in the model and the actual measurements of the object or area.
5. Population Ratios: Population ratios compare various population groups. For example, the ratio of males to females in a town might be 3:2, meaning there are three males for every two females.

Ratio Problems

Example 1:
The ratio of boys to girls in a class is 3:5. If there are 48 students in total, how many boys are there?

To solve this problem, we can set up a proportion. Let x represent the number of boys.

• Boys: Girls = 3:5
• Boys + Girls = Total students

Writing this as a proportion, we get: 3/5 = x/(48-x)

To solve for x, we can cross-multiply: 3(48-x) = 5x
Simplifying the equation, we get: 144 - 3x = 5x
Rearranging the equation, we have: 8x = 144
Dividing both sides by 8, we find: x = 18

Therefore, there are 18 boys in the class.

Example 2:
A recipe calls for a ratio of 2 cups of flour to 1 cup of milk. If you want to make 6 cups of flour, how much milk do you need?

To solve this problem, we can set up a proportion. Let x represent the amount of milk needed.

• Flour: Milk = 2:1

Writing this as a proportion, we get: 2/1 = 6/x

To solve for x, we can cross-multiply: 2x = 6
Dividing both sides by 2, we find: x = 3

Therefore, you would need 3 cups of milk.

Proportional Problems

Sure! Here are a few examples of proportional problems:

Example 1:
A car can travel 200 miles on 8 gallons of gas. How far can it travel on 4 gallons of gas?

To solve this problem, we can set up a proportion. Let x represent the distance the car can travel on 4 gallons of gas.

• Distance: Gas = 200:8

Writing this as a proportion, we get: 200/8 = x/4

To solve for x, we can cross-multiply: 8x = 800
Dividing both sides by 8, we find: x = 100

Therefore, the car can travel 100 miles on 4 gallons of gas.

Example 2:
A scale model of a building has a height of 12 inches. If the actual building has a height of 60 feet, what is the scale ratio of the model to the actual building?

To solve this problem, we can set up a proportion. Let x represent the scale ratio of the model to the actual building.

• Model Height: Actual Height = 12:60

Writing this as a proportion, we get: 12/60 = x/1

To solve for x, we can cross-multiply: 60x = 12
Dividing both sides by 60, we find: x = 0.2

Therefore, the scale ratio of the model to the actual building is 1:0.2.

New Jersey is a state located in the Mid-Atlantic region of the United States. It is bordered by New York to the north and northeast, Pennsylvania to the west, and Delaware to the southwest.

To the east, New Jersey is bordered by the Atlantic Ocean. Its capital is Trenton, and its largest city is Newark.

Known as the Garden State, New Jersey is known for its beautiful beaches, diverse culture, and proximity to major cities like New York City and Philadelphia.

New Jersey is the most densely populated state in the United States, with a population of over 9 million people. The state capital is Trenton, and the largest city is Newark. Other major cities include Jersey City, Paterson, Elizabeth, and Camden.

New Jersey is known for its beaches, casinos, and farms. The state is also home to many Fortune 500 companies, including Johnson & Johnson, Merck, and Prudential Financial.

Here are some fun facts about New Jersey:

• The state is known as the “Garden State” because of its many farms.
• New Jersey is home to the Liberty Bell, which was a symbol of the American Revolution.
• The state is also home to Ellis Island, which was the gateway for millions of immigrants to the United States.
• New Jersey is the birthplace of Frank Sinatra, Bruce Springsteen, and Whitney Houston.

Highest Common Factor

The highest common factor, also known as the greatest common divisor, is the largest positive integer that divides two or more numbers without leaving a remainder. It is commonly denoted as HCF or GCD.

To find the highest common factor of two numbers, you can use various methods such as prime factorization, division method, or the Euclidean algorithm.

For example, let’s find the highest common factor of 24 and 36 using the prime factorization method:

Prime factorization

1. Find the prime factors of each number:

   • 24 = 2 * 2 * 2 * 3 = 2^3 * 3
   • 36 = 2 * 2 * 3 * 3 = 2^2 * 3^2

2. Identify the common prime factors and their lowest powers:

   • Common factors: 2^2 * 3

3. Multiply the common factors to find the highest common factor:

   • HCF(24, 36) = 2^2 * 3 = 12

Therefore, the highest common factor of 24 and 36 is 12.

Finding the highest common factor is important in various mathematical operations, simplifying fractions, and solving mathematical problems.

Division Method

To find the highest common factor (HCF) of 24 and 36 using the division method, follow these steps:

1. Start by dividing the larger number (36) by the smaller number (24):

   36 ÷ 24 = 1, with a remainder of 12

2. Now divide the divisor (24) by the remainder (12):

   24 ÷ 12 = 2

3. Divide the remainder from the previous step (12) by the new remainder (0):

   12 ÷ 0 = undefined

Since the remainder has become zero, the HCF is the divisor from the last non-zero remainder. In this case, the HCF of 24 and 36 using the division method is 12.

Therefore, the highest common factor of 24 and 36 is 12 when using the division method.

Lowest Common Multiple

The Lowest Common Multiple (LCM) is the smallest positive integer that is divisible by two or more numbers. To find the LCM of two or more numbers, you can follow these steps:

1. Determine the prime factors of each number involved.
2. Identify the highest power of each prime factor that appears in any of the numbers.
3. Multiply together all the highest powers identified in the previous step.

For example, let’s find the LCM of 9 and 12:

Step 1: Prime factors of 9 are 3 x 3 (3^2), while prime factors of 12 are 2 x 2 x 3 (2^2 x 3).

Step 2: The highest power of 2 is 2^2, and the highest power of 3 is 3^2.

Step 3: Multiply the highest powers: 2^2 x 3^2 = 4 x 9 = 36.

Therefore, the LCM of 9 and 12 is 36.

Remember, you can apply the same steps to find the LCM of more than two numbers.

LCM Examples

1. Vulva: The female reproductive structures that lie external to the va-gina are called the external genitalia or vulva comprising labia majora, labia minora, hymen and cli-toris.

2. Labia majora: Labia Majora means large lips. It is large, fleshly folds of tissue that protect and enclose other external genital organs. Labia majora contains sebaceous and sweat glands, these glands produce lubricating secretion.

3. Labia minora: Labia Minora means minor lips. These are paired folds of tissue that underlie the labia majora. These tissues cover the urethral and va-ginal opening, during sexually unstimulated conditions.

But Labia minora becomes more open during sexual arousal.

4. Vestibule:

The vestibule contains numerous sensory nerve endings, which play a role in sexual pleasure and sensitivity.

The vestibule, along with the labia minora and majora, collectively forms the external genitalia of the female reproductive system. It is an important and highly sensitive area involved in sexual function, urination, and lubrication, and it plays a crucial role in female reproductive health and sexuality.

5. Cli-toris: (with 6. Glans and 7. Body).

8. Bulb of the vestibule
9. Va-gina:

10. Hymen;

11. Lumen;

12. Wall;

13. Fornix (lateral)
14. Uterus: Parts:

15. Cervix;

16. Body and 17. Fundus.

18. Orifices: External and Internal;

19. Cervical canal;

20. Uterine cavity; Layers:

21. Endometrium;

22. Myometrium

23. Perimetrium
24. Fallopian tube:

25. Isthmus;

26. Ampulla;

27. Infundibulum;

28. Fimbria (with 29. Fimbria ovarica)
30. Ovary
31. Visceral pelvic peritoneum:

32. Broad ligament (with 33. Mesosalpinx; 34. Mesovarium and 35. Mesometrium)
Ligaments:

36. Round;

37. Ovarian;

38. Suspensory of the ovary
Blood vessels:

39. Ovarian artery and vein;

40. Uterine artery and veins;

41. Va-ginal artery and veins
Other:

42. Ureter;

43. Pelvic floor (Levator ani);

44. Femoral head;

45. Hip bone;

46. Internal iliac vessels (anterior branches);

47. External iliac vessels;

48. Abdominal cavity

The female reproductive system is far more complex than the male’s because, in addition
to gamete formation, it has to nurture the developing foetus. The female reproductive
system consists of a pair of ovaries along with a pair of oviducts, uterus, cervix, va-gina and
external genitalia located in the pelvic region.

These parts along with the mammary glands are integrated structurally and functionally to support the process of ovulation, fertilisation, pregnancy, childbirth and child care. Ovaries are the primary female sex organs that produce the female gamete, the ovum.

The ovaries are located on each side of the lower abdomen. The ovary is an elliptical structure
about 2-4 cm long. Each ovary is covered by a thin cuboidal epithelium called the germinal
epithelium which encloses the ovarian stroma.

The stroma is differentiated into the outer cortex and inner medulla. Below the germinal epithelium is
a dense connective tissue, the tunica albuginea. The cortex appears dense and granular due
to the presence of ovarian follicles in various stages of development.

The medulla is a loose connective tissue with abundant blood vessels, lymphatic vessels and nerve fibres. The ovary remains attached to the pelvic wall and the uterus by an ovarian ligament
called mesovarium.

The fallopian tubes (uterine tubes or oviducts), uterus and va-gina constitute the female accessory organs. Each fallopian tube extends from the periphery of each ovary to the uterus. The proximal
part of the fallopian tube bears a funnel-shaped infundibulum.

The edges of the infundibulum have many finger-like projections called fimbriae which help in the collection of the ovum after ovulation.

The infundibulum leads to a wider central portion called the ampulla. The last part of the
the oviduct is the isthmus which is short and thick-walled connecting the ampulla and
infundibulum to the uterus.

The uterus or womb is a hollow, thick-walled, muscular, highly vascular and inverted pear-shaped structure lying in the pelvic cavity between the urinary bladder and rectum. The major portion of the uterus is the body and the rounded region superior to it, is the fundus.

The uterus opens into the va-gina through a narrow cervix. The cavity of the cervix is called the
cervical canal communicates with the va-gina through the external orifice and with the uterus
through the internal orifice.

The cervical canal along with the va-gina forms the birth canal. The wall of the uterus has three layers of tissues. The outermost thin membranous serous layer is called the perimetrium, the middle thick muscular layer is called the myometrium and the inner glandular layer is called the endometrium.

The endometrium undergoes cyclic changes during the menstrual cycle while the myometrium exhibits strong contractions during parturition. The va-gina is a large fibromuscular tube that extends from the cervix to the exterior.

It is the female organ of copulation.

The female reproductive structures that lie external to
the va-gina are called the external genitalia or vulva comprising labia majora, labia minora, hymen and cli-toris.

The Bartholin’s glands (also called greater vestibular glands) are located posterior to the left and right of the opening of the va-gina. They secrete mucus to lubricate the va-gina and are homologous to the
bulbourethral glands of the male.

The Skene’s glands are located on the anterior wall of the va-gina and around the lower end of the urethra. They secrete a lubricating fluid and are homologous to the prostate gland of males. The external opening of the va-gina is partially closed by a thin ring of tissue called the hymen.

The hymen is often torn during the first coitus (physical union). However, in some women, it remains intact. It can be stretched or torn due to a sudden fall or jolt and also during strenuous physical activities such as cycling, horseback riding, etc., and therefore cannot be considered an indicator
of a woman’s virginity.

Letter σ. The standard deviation is represented in the Greek letter σ (sigma). This term refers to the amount of variability in a given set of data.

The population standard deviation, denoted by σ, is a measure of how spread out the values of a population are from the population mean. It is calculated using the following formula:

σ = √(∑(x_i - μ)^2 / N)

where:

• σ is the population standard deviation
• x_i is each value in the population
• μ is the population mean
• N is the number of values in the population

To calculate the population standard deviation, you would first need to calculate the population mean. Then, you would subtract the mean from each value in the population to get the deviations. Next, you would square each deviation to make it positive. Finally, you would add up all of the squared deviations and divide by the number of values in the population. The square root of this result is the population standard deviation.

For example, let’s say we have the following population of heights:

150, 155, 160, 165, 170

To calculate the population standard deviation, we would first need to calculate the population mean. The population mean is calculated by adding up all of the values in the population and dividing by the number of values. In this case, the population mean is 160.

Next, we would subtract the mean from each value in the population to get the deviations. The deviations are:

-10, -5, 0, 5, 10

Now, we would square each deviation to make it positive:

100, 25, 0, 25, 100

Next, we would add up all of the squared deviations:

250

Finally, we would divide the sum of the squared deviations by the number of values in the population and take the square root to get the population standard deviation. The population standard deviation is 10.

A population standard deviation of 10 means that the average height in the population is 160, and that the typical height is within 10 units of the mean.

It is important to note that the population standard deviation is a population parameter, meaning that it is a measure of the variability of the entire population. In practice, it is often difficult or impossible to calculate the population standard deviation, because we do not have data for the entire population. Instead, we often estimate the population standard deviation by calculating the sample standard deviation.

Rajaji National Park is an Indian National Park and Tiger Reserve. It is situated in the region of Shivaliks near the Himalaya foothills. It spans over 820 Km and includes the Haridwar, Dehradun, and Pauri Garhwal districts of Uttarakhand State of India.

I was angry with my friend;
I told my wrath, my wrath did end.
I was angry with my foe:
I told it not, my wrath did grow.
And I water’d it in fears,
Night and morning with my tears:
And I sunned it with smiles,
And with soft deceitful wiles.
And it grew both day and night,
Till it bore an apple bright .
And my foe beheld it shine,
And he knew that it was mine.
And into my garden stole,
When the night had veiled the pole;
In the morning glad I see,
My foe outstretched beneath the tree.