6th Class Mathematics Ratio and Proportion Ratio, Proportion & Unitary Method

RATIO, PROPORTION AND UNITARY METHOD

RATIO

The comparison of two quantity of same kind by division is called ratio.

Example: Ratio between Rs. 30 and Rs. 50, but there can be no ratio between Rs. 30 and 50 apples, form of ratio \[=x:y\]

\[x\to \]antecedent

\[y\to \]consequent  

•           B The ratio of \[x\]to \[y\]
•           B \[x\]is to \[y\]
•           \[x:y\]
•           B \[x\]and \[y\] are called terms of ratio.

Types of Ratio:

•                Compound Ratio: Ratio is compound when antecedents are multiplied by respective antecedents and consequents are multiplied by respective consequents.

Example: \[a:b,\,\,c:d,\,\,e:f,\]then compound ratio is,\[\frac{a\times c\times e}{b\times d\times f}\]

•                   Duplicate Ratio: If \[x:y\] is a ratio, then\[{{x}^{2}}:{{y}^{2}}\]is duplicate ratio.

Example: Find duplicate ratio of\[5:7.\]

Solution: \[{{\left( 5 \right)}^{2}}:{{\left( 7 \right)}^{2}}=25:49\]

•                   Triplicate Ratio:  If \[x:y\] is a ratio, then \[{{x}^{3}}:{{y}^{3}}\]is

•                 Triplicate Ratio.

Example: Find triplicate ratio of\[2:3\].

Solution: \[{{\left( 2 \right)}^{2}}:{{\left( 3 \right)}^{2}}=8:27\]

•                   Sub-duplicate ratio: If \[x:y\] is a ratio then\[\sqrt{x}:\sqrt{y}\] is sub-duplicate ratio.                       

Solution: \[\sqrt{4}:\sqrt{9}\]

\[=\sqrt{2\times 2}:\sqrt{3\times 3}\]\[=2:3\]

•                  Sub-triplicate ratio: If \[x:y\] is a ratio then \[\sqrt[3]{x}:\sqrt[3]{y}\] is sub-triplicate ratio.           

Example: Find sub-triplicate ratio of\[1:8\].

Solution:\[\sqrt[3]{1}:\sqrt[3]{8}=1:2\]

Proportion;

•                  The Equality of two ratio is called proportion.
•                  \[w:x::\text{ }y:z\]then \[w,\,\,x,\,\,y\] and \[z\]are said to be in proportion. Then \[w,\,\,x,\,\,y\] and \[z\] are called terms.
•                   \[:\,\,:\] sign of proportion.
•                The first and fourth i.e. \[w\] and \[z\] are called extremes and the second and third terms i.e. \[x\]and \[y\] are called means, z is called fourth proportional.
•                   Product of extremes = product of means

\[w\times z=x\times y\]

\[wz=xy\]

Example: Let the four quantities 5, 10, 6 and 12 be in proportion.

\[\frac{5}{10}=\frac{6}{12}\]or  \[5:10::6:12\]

•                 These quantities of the same kind are said to be in 'continued' proportion when the ratio of the first to the second is equal to the ratio of the second to the third.
•                  The second quantity is called mean proportional between the first and the third and the third quantity is called third proportional to the first and the second.

Example: 9, 6 and 4 are in continued proportion for\[9:6::6:4\].

Hence, 6 is the mean proportional between 9 and 4, and 4 is called third proportional to 9 and 6.

Mean Proportion

•                  \[x:y::y:z\Leftrightarrow {{y}^{2}}=xz\]

Example:     Find the fourth prepositional to the number 6, 8 and 15.

Let \[x\] be fourth proportional

\[\therefore \]\[6:8::15:x\]

\[6x=8\times 15\]

\[x=\frac{120}{6}=20\]

Unitary Methods

•                   Unitary method is a way of calculating the value of a number of items by first finding the cost of one of them.
•                   Value of one item\[\text{=}\frac{\text{Value of given quantity of items}}{\text{Quantity of items}\text{.}}\]
•                   Value of required quantity = value of one item \[\times \] Required quantity of items

Example: If the cost of 4 pencils is Rs. 24. What will be the cost 7 pencils.

Solution: We have,

Cost of 4 pencils = Rs. 24

\[\therefore \]  Cost of 1 pencil\[=Rs.\frac{24}{4}\]

•                  Hence, cost of 7 pencils

\[=Rs.\left( \frac{24}{4}\times 7 \right)=Rs.\left( 6\times 7 \right)=Rs.42\]