Highest Common Factor (H.C.F.):
The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.
Methods of finding the H.C.F. of a given set of numbers:
Factorization Method: Expression each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.
Division Method: Suppose we have to find the H.C.F. of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required H.C.F.
Finding the H.C.F. of more than two numbers:
Suppose we have to find the H.C.F. of three numbers then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers.
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The ratio of two quantities ‘a’ and ‘b’ in the same unit, is the fraction a/b and it is denoted as a: b.
The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Example: 4: 5 = 8: 10 = 12: 15 etc.
Proportion:
The equality of two ratios is called proportion.
If a : b = c : d, also write as a : b : : c : d . ‘a’ and ‘d’ are called extremes, while ‘b’ and ‘c’ are called mean terms.
Product of means = Product of extremes.
Thus, a : b : : c : d Û (b × c) = (a × d).
Points to remember:
Read More...If A can do a piece of work in n days, then A’s 1 day’s work = 1/n. The same concept is applied in case of pipes and cisterns. If a pipe takes n hours to fill a tank then in 1 hur it will fill 1/n part of tank and if a pipe can empty the tank in n hours then in 1 hour it will empty 1/n part of tank.
Inlet: A pipe connected with a tank or a cistern or a reservoir, that fills it, is known as an inlet.
Outlet: A pipe connected with a tank or a cistern or a reservoir, emptying it, is known as outlet.
If one pipe can fill tank in a hours and other can empty it in b hours then in 1 hour the tank will be filled or empty depends upon the value of a and b.
(1/a – 1/b ). If the calculation is –ve then tank will be empty and if calculation is +ve then the tank will be filled up.
If A is thrice as good a workman as B, then:
Ratio of work done by A and B = 3 : 1.
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