Young mathematicians

1. Ratio and proportion

1 .In the ratio a:b; a,b are called terms of the ratio. a is called the first term and b is called the second term.If two quantities expressed in the same units are compared by division the result obtained is called a ration.
2. If we multiply or divide the ratio by a non-zero number its value does not change.
3. In a ratio either the first term or the second term be a fraction take the L.C.M of the denominators and multiply with the denominators to the given ratio,then they are changed into a whole numbers. Ratios are always represented in whole numbers.
4. For putting into simplest form we must divide the ratio by the G.C.F of the terms.
5. there are no units for a ratio.
Example: If a:b =6:5; b:c=4:1 the find a:b:c?
sol. b is the common terms in the given ratio and they are not equal.
        L.C.M of 5 and 4 is 20.
        a:b = 6:5 = 6 x 4 : 5 x 4 = 24 : 20.
        b:c = 4:1 = 4 x 5 : 1 x 5 = 20 : 5
       Therefore, a : b : c = 24 : 20 : 5
 The inverse ratio of a:b is b:a
 The duplicate ratio of a:b is

If two ratios are equal then the terms in the given ratios are said to be in proportion.
If a:b = c:d then a,b,c,d are in proportion.It is denoted by a/b=c/d or a:b : : c:d     
  :: This symbol is read as"is as".
      a,d are called extremes.
      b,c are called means.
If four numbers are in proportion ,then  The product of "Extremes" =  The product of "means"
F a:b= c:d then ad= bc,
If they are not in proportion  the product of extremes ≠ the product of means.
If a,b,c are in proportion b is called the mean proportional of a and c and b square = ac
Example: Find the fourth no. 16,12,4 are in proportion.
sol. Let the fourth number be x
Therefore 16,12,4,x are in proportion.
Product of the extremes = The product of the means.
          16 X x = 12 x 4
           x = 12 x 4 / 16
The fourth number = 3.
Example 2:Find the mean proportion of 16,9?
sol. Mean proportion = √16 x 9 = √ 144 = 12.

Direct proportion or Direct variation

If two quantities are so related than an increase or decrease in one causes a corresponding increase or decrease in other then they are said to be in Direct proportion or Direct variation.
If in two variables x and y, x/y is constant , then x and y are said to be in Direct variation.
We read this as "x is directly proportional to y."

Inverse proportion or Inverse variation

If  two quantities are so related that an increase or decrease in one causes a corresponding decrease or increase in the other then they are said to be in an inverse proportion or inverse variation.
      If two quantities x and y change such that  x X y = k ( constant ) then x and y are said to be in inverse variation.
     In this relation x X y = k   one of the variable increases or decreases their product must be constant so the second variable decreases or increases.
               x,y are in inverse variation then x X y = k (constant this can be represented x/1/y = k or y/1/x = k)
             x, 1/y and y, 1/x are in proportion.
         x, y are inverse proportion then x α1/y and y α 1/x.
 x, y are in inverse proportion it can be denoted by x α y and it is read as "x is inversely proportional to y."


1. Fractions having denominators as 100 are called Percentages.Percentage is denoted by the symbol %.
2. To convert a common fraction into % we must multiply the fraction with 100 and put % symbol to the result.
3. To convert a   %   into a fraction ,we must remove % symbol, multiply with 1/100 and simplify the product.
4. If there is an increase of x% in a quantity then its value increases by (100+x)over its original value . multiplying fraction to get the increased value is 100+x/100.
5. If there is a decrease of x% in a quantity the its value decreases by (100-x)%t over its original value.multiplying fraction to get the decreased value is 100-x/100.
Example : Ravi deposited Rs.6500 in a bank which pays 9% interest per anum .How much interest will he get at the end of the year?

sol. Interest paid the bank = 9% = 9/100      
       Therefore Amount of interest Ravi  gets at the end of the year
                       = 9/100 of Rs.6500=9/100x6500=Rs.585

Profit and loss

 The money paid by shopkeeper to buy goods is called cost price of the shop keeper.
   The price at which he sells the goods is called the Selling price of the shopkeeper.
   If the selling price is greater than the cost price then the shopkeeper gets profit.
   If the selling price is less than the cost price then the shopkeeper gets loss.
   Loss = C.p - S.p.
   Profit = S.p +Loss.
   % of Profit = Profitx 100
   % of Loss = Loss  x 100
   Cost price =  S.p x 100
                       100 + profit %.
   Cost price = S.p x 100
                       100 - loss %.
   Selling price = c.p(100 + gain % )
   Selling price = c.p( 100 - loss % )