Arithmetic help (show steps).

Question

1. What is the no. of people among whom 125 bananas and 145 orange can be equally distributed?
2. What is H.C.F? Show any example how to calculate it.
3. The difference of the numerator and the denominator of a proper fraction is 2. When the fraction obtained by subtracting 3 from both numerator and denominator is added to 1/4, the result is 1. What is the fraction?
4. The sum of the numerator and denominator of a fraction is 11, and the fraction formed by subtracting 2 from the numerator and adding 3 to the denominator is 1/2. Find the fraction.

Answer 1

1) GCF of 125 and 145 :

Per Euclid's algorithm, divides:

145 / 125 = 1 remainder 20

125 / 20 = 6 remainder 5

20 / 5 = 4 remainder 0

the last divisor is 5, so the GCF (125,145) = 5

125/5 = 25 bananas and 145/5 = 29 oranges each

2) There are two ways to find the GREATEST COMMON FACTOR (GCF)

or as you call it the HIGHEST COMMON FACTOR (HCF) , same thing

method 1: factor trees;

145 = 5 * 29

125 = 5 * 25 = 5*5*5

the factors in common is 5, so 5 is the GCF

Here is another example: GCF( 32, 48)

32 = 2*2*2*2*2

48 = 2*2*2*2*3

the common factors are 2*2*2*2 = 16, so the GCF=HCF=16

method 2: Euclid's ALgotrithm

Step 1: Divides larger by smaller, and write the remainder

Step 2: divides the smaller of the two number by the remainder

Step 3: Repeat step 2 until the remainder is zero (0)

step 4: the last DIVISOR is the GCF

Euclid's algorithm was demonstrated in (1).

For the GCF(32,48), Euclid's algorithm says:

48/32 = 1 remainder 16

32 / 16 = 2 remainder 0

the last divisor is 16, so GCF(32,48)=16

(3) N is the numerator, so then the fraction

is N/N+2

(N-3)/(N-1) + 1/4 = 1/1 = 1

[4(N-3) + (N-1) ]/ [4(N-1)] = 1

[ 4n-12 +n-1] / [4n-4] = 1

[ 5n - 13]/ [4n-4] = 1

5n-13 = 4n-4

n - 13 = -4

n = 9

the original fraction is 9/11

subtracts three from numerator and denominator: 6/8 = 3/4

adding 1/4 gives 3/4+1/4 = 1

so yes, the original fraction is 9/11

(4) N is the numerator. So the fraction is n/(11-n)

Then (n-2) / (14-n) = 1/2

cross multiplies: 2(n-2) = 14-n

2n - 4 = 14-n

3n - 4 = 14

3n = 18

n=6

the original fraction is 6/5

subtracting 2 from the numerator and adding 3 to the denominator

makes the fraction 4/8 = 1/2