solve yhe problems based on alligation(for BBA 1st)

1.	A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
	A. 1 3 B. 1 4 C. 1 5 D. 1 7

2.  Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1 : 1 : 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be: A. Rs. 169.50 B. Rs. 170 C. Rs. 175.50 D. Rs. 180                       3.  A can contains a mixture of two liquids A and B is the ratio 7 : 5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. How many litres of liquid A was contained by the can initially? A. 10 B. 20 C. 21 D. 25   4.  A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3 : 5? A. 4 litres, 8 litres B. 6 litres, 6 litres C. 5 litres, 7 litres D. 7 litres, 5 litres

unit 1 ratio & mixture

            ALLIGATION OR MIXTURES

Important Facts and Formula:

1.Allegation:It is the rule that enables us to find the

ratio in which two of more

ingredients at the given price must be

mixed to produce a mixture of a desired price.

2.Mean Price:The cost price of a unit quantity of the mixture

is called the mean price.

3.Rule of Allegation:If two ingredients are mixed then

          Quantity of Cheaper / Quantity of Dearer =
                             (C.P of Dearer – Mean Price) /(Mean Price–C.P of Cheaper).

                           C.P of a unit quantity of cheaper(c)
                           C.P of unit quantity of dearer(d)
                           Mean Price(m)
                               
                          (d-m) (m-c)

 Cheaper quantity:Dearer quantity = (d-m):(m-c)
 
4.Suppose a container contains x units of liquid from which y units
are taken out and replaced by water. After n operations the
quantity of pure liquid = x (1 – y/x)n units.
 
SOLVED PROBLEMS

Simple problems:
1.In what ratio must rice at Rs 9.30 per Kg be mixed with rice
at Rs 10.80 per Kg so that the mixture be worth Rs 10 per Kg?
Solution:
C.P of 1 Kg rice of 1st kind 930 p C.P of 1 Kg rice of 2n d kind 1080p
Mean Price 1000p
80 70

unit 1 ratio & proportion

Ratio:

A ratio is simply a fraction.   The following notations all express the ratio of x to y
=>  $x:y$ , $x÷y$ , or $\frac{x}{y}$.
In the ratio x : y, we call a as the first term or antecedent and b, the second term or consequent.
Writing two numbers as a ratio provides a convenient way to compare their sizes.   For example, since $\frac{3}{\pi }<1$, we know that 3 is less than π.
A ratio compares two numbers.  Just as you cannot compare apples and oranges, so the numbers you are comparing must have the same units. 
For example, you cannot form the ratio of 2 feet to 4 meters because the two numbers are expressed in different units—feet vs. meters.

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Example 1: What is the ratio of 2 feet to 4 yards?
(A)  1 : 2        (B)  1 : 8          (C)  1 : 7        (D)  1 : 6            (E)  1 : 5       
The ratio cannot be formed until the numbers are expressed in the same units.  Let’s turn the yards into feet.
Since there are 3 feet in a yard, 4 yards = 4 * 3 feet = 12 feet . 
Forming the ratio yields $\frac{\text{2 feet}}{\text{12 feet}}=\frac{1}{6}$ or $1:6$
The answer is (D).
Note. Taking the reciprocal of a fraction usually changes its size.  For example,
$\frac{3}{4}\ne \frac{4}{3}$
So order is important in a ratio=> 3:4 ≠  4:3.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Ex. 4 : 5 = 8 : 10 = 12 : 15.
Also, 4 : 6 = 2 : 3

Proportion:

The equality of two ratios (fractions) is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
$$\text{Product of means = Product of extremes}$$
Thus, $$a:b::c:d\iff \left(b*c\right)=\left(a*d\right)$$

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Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.

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Third Proportional:

a : b = c : d, then c is called the third proportional to a and b.

Mean Proportional:

Mean proportional between a and b is ab.

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Comparison of Ratios:

We say that $\left(a:b\right)>\left(c:d\right)\iff \frac{a}{b}>\frac{c}{d}$

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Compounded Ratio:

The compounded ratio of the ratios: $\left(a:b\right),\left(c:d\right),\left(e:f\right)is\left(ace:bdf\right)$

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Duplicate Ratios:

Duplicate ratio of $\left(a:b\right)$ is $\left(a^2:b^2\right)$
Sub-duplicate ratio of $\left(a:b\right)$ is $\left(a^{\frac{1}{2}}:b^{\frac{1}{2}}\right)$
Triplicate ratio of $\left(a:b\right)$ is $\left(a^3:b^3\right)$
Sub-triplicate ratio of $\left(a:b\right)$ is $\left(a^{\frac{1}{3}}:b^{\frac{1}{3}}\right)$
If $\frac{a}{b}=\frac{c}{d}$ then, $\frac{a+b}{a-b}=\frac{c+d}{c–d}$ [Componendo and Dividendo]

Variations:

We say that x is directly proportional to y,
if x = ky for some constant k and we write, $x\propto y$
We say that x is inversely proportional to y,
if xy = k for some constant k and we write, x∝1

unit 1 some imp points about ratio

Definition:

A ratio is a comparison of two similar quantities obtained by dividing one quantity by the other. Ratios are written with the : symbol.

Example:The ratio of 6 to 3 is 6 ÷ 3 = 6/3 = 6 : 3 = 2

Example:The ratio of 3 to 6 is

3 ÷ 6 = 3/6 = 3 : 6 = 1/2

Notes about ratios:

Since a ratio is only a comparison or relation between quantities, it is an abstract number. For instance, the ratio of 6 miles to 3 miles is only 2, not 2 miles.

As you can see above, ratios can be written as fractions. They also have all the properties of fractions that you have learned in the previous part of this station.

The ratio of 6 to 3 should be stated as 2 to 1, but common usage has shortened the expression of ratios to be called simply 2.

If two quantities cannot be expressed in terms of the same unit, there cannot be a ratio between them.

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Can you find these ratios?

1.2 quarts to 7 gallons

2.5 dollars to 25 cents

3.4 meters to 3 kilograms

Problem:

If two full time employees accomplish 20 tasks in a week, how many such tasks will 5 employees accomplish in a week?

2 : 5 = 20 : x

2 × x = 5 × 20

x = 50 tasks

This answer is obtained by knowing about proportions and how they are used. You can set up proportions by using ratios. Remember, ratios are comparing similar things. In the problem above, the first ratio is comparing employees and the second is comparing tasks.

Definition:

A proportion is a statement of the equality of two ratios.

Example:6 : 3 = 2 : 1 or 6 / 3 = 2 / 1or

62
- = -
31

are ways to write the proportion expressed as: 6 is to 3 as 2 is to 1

Example:2 : 8 = 1 : 4 or 2 / 8 = 1 / 4or

21
- = -
84

are ways to write the proportion expressed as: 2 is to 8 as 1 is to 4

Notes about proportions:

If any three terms in a proportion are given, the fourth may be found. Given the proportion:

a : b = c : d or a / b = c / d

and using the principals of manipulating equations , we observe that

a = (b × c) / d and c = (a × d) / b

b = (a × d) / c and d = (b × c) / a

An easy way to remember this is to say that in a proportion:

The product of the means is equal to the product of the extremes.

It is important to remember that to use the proportion, the ratios must be equal to each other and must remain constant.

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Try a few proportion problems:

1.Find the value of x in 16 : 8 = x : 5

2.Find the value of x in 25 : 15 = 10 : x

3.A pipe transfers 236 gallons of fuel to the tank of a ship in 2 hours. How long will it take to fill the tank of the ship that holds 4543 gallons?

4.An I-beam 12 feet long weighs 52 pounds. How much does an I-beam of the same width weigh if it is 18 feet long?