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Ratios and Proportions

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A Ratio is a comparison of two similar quantities obtained by dividing one quantity by the other. The quantities to be compared should be of the same unit. Two similar quantities when compared, will not give any unit. The ratio is written with " : " symbol. The ratio of two quantities are written as, a : b. It can also be written as $\frac{a}{b}$.

When two ratios are equal, they are called Proportions. The proportion is written as a : b = c : d. The cross product of the two ratios can also be equalized a x d = b x c. If three values in a proportion are known, then we can easily find the fourth value. For the proportion to be used, the ratios must be equal.

    Ratio Definition

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    A ratio is a relationship between two numbers of the same kind. Usually expressed as A : B or $\frac{A}{B}$ or by the phrase "A to B", where, 'A' indicates the quotient of two mathematical expressions and 'B' is the relationship that can be in quantity, amount or size between two or more things.

    It is a statement of how two numbers are compared. It is the comparison of the size of one number to the size of another number.
    In layman terms, a ratio represents, for every amount of one thing, how much there is of another thing.

    Proportion Definition

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    Proportion is a name given to a statement, when the two ratios are equal. It can be written in two ways as follows
    Two equal fractions, $\frac{a}{b}$ = $\frac{c}{d}$
    (or)
    Using a colon, a : b = c : d

    In a proportion, two ratios are equal and if the things are in proportion, then their relative sizes are same.
    For example: 1 strawberry : 5 oranges
    This ratio compares strawberries to oranges. It means that for every strawberry, there are 5 oranges.

    Ratio and Proportion Problems

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    Given below are some of the problems on ratio and proportions.

    Solved Examples

    Question 1: Find the value of x in the given problem
    4 : 3 = x : 9
    Solution:
    We can write 4 : 3 = x : 9 as $\frac{4}{3}$ = $\frac{x}{9}$
    4 $\times$ 9 = x $\times$ 3
    x $\times$ 3 = 4 $\times$ 9
    x = $\frac{4 \times 9}{3}$
    x = 12

    Question 2: What is the ratio of 8 minutes to 9 hours?
    Solution:
    Change the hours to minutes.
    9 hours = 9 $\times$ 60 = 540 minutes

    The ratio is then to be written in terms of fraction and simplified.
    $\frac{8}{540}$ = $\frac{1}{67.5}$

    Therefore, the ratio of 8 minutes to 9 hours is 1 : 67.5

    Question 3: Find the unknown value in the proposition (7x + 2) : 4 = (x + 5) : 2
    Solution:
    In terms of fraction, the given equation can be written as

    $\frac{7x+2}{4}$ = $\frac{(x+5)}{2}$

    Solve the fraction to find the value of x

    2(7x + 2) = 4(x + 5)
    14x + 4 = 4x + 20
    10x = 16
    x = $\frac{8}{5}$

    How to Solve Ratios?

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    Given below are the basic steps to be remembered while solving ratios:
    1. Check whether the quantities are in the same units, then reduce it to its simplest form.
    2. Write the items in the ratio form.
    3. In ratio, the items should be same in the numerator as well as in the denominator.

    How to Solve Proportions?

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    Given below are the three methods used to solve proportions:
    Vertical:

    Let us check whether the given ratios are equivalent.

    $\frac{6}{12}$ and $\frac{5}{10}$
    How to Solve Proportions
    As you can see ,the above ratios are equivalent.
    The above two ratios are equivalent. Numerator and denominator are related.

    Horizontal:
    Let us check whether the given are ratios equivalent.

    $\frac{6}{7}$ and $\frac{18}{21}$

    How to Solve Proportion
    As you can see, the above ratios are equivalent.
    As the numerator and denominator are related to each other, the ratios are equivalent.

    Cross Products
    Let us check whether the two ratios are equivalent.
    $\frac{4}{6}$ and $\frac{16}{24}$

    Solve Proportions
    As you can see, the above ratios are equivalent above as you can see.
    Cross products are equal to each other.
    So, the ratios are equivalent.