A person went to a super market and purchased some items and calculated the bill. After that he added few more items to the list and observed that there is an increasing of the total cost in the latest bill. Here the total cost is getting increased with the increase of the total number of items. Here we say the *total cost* and the *number of items* are in the direct proportion.

Mathematically the ** direct proportion** is defined as follows.

Two variables ** a** and

**are said to be in the direct proportion if both of them increase (or decrease) together such that the ratio of corresponding values remains constant. i.e.**

*b***and**

*a***are in direction proportion in only the following cases.**

*b*i) a increases when b increases

ii) b decreases when a decreases.

So, if ** a** and

**are in direct proportion their ratio is constant.**

*b*i.e., a/b = k (where ** k** is a constant)

i.e. *a=bk*

If ** a_{1}** and

**are any two values of**

*a*_{2}**and**

*a***and**

*b*_{1}**are any two values of**

*b*_{2}**then**

*b*a1/b1 = k and a2/b2 =k

From these two equations, we get

*a1/b1=a2/b2*

This relation is mostly used in solving the mathematical problems with direct proportion.

**Example:**

A paint is prepared using 1 part of red pigments with 80 ml of base, then how much red pigment is to be mixed with 1760 ml of base?

**Solution:**

Here, we need to observe that more the base requires more parts of red pigments.

So, we can say that red pigments and the base are in the direct proportion.

Red pigment (a) | Base (b) |

1 (a1) | 80 ml (b1) |

a2 | 1760 ml (b2) |

By the definition of direct proportion,

*a1/b1=a2/b2*

1/80 = a_{2}/1760

Multiplying both sides by 1760, we get

a_{2} = 1/80 *1760 = 22.

So, we need 22 parts of red pigments to mix with 1760 ml of base.

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