** L.C.M and H.C.F of fractions**

Just like we found for whole numbers, we can also calculate the L.C.M and H.C.F of fractions. Such as ½, ¾, 0.3, ¼, 2.1 etc.

L.C.M =

H.C.F =

When decimal numbers are involved, change the decimal numbers to fractions and proceed to find the lcm and hcf with the above formulas.

** Example**

1. Find the lcm of 5/8, 3/4 and 9/20.

__Solution__

Lcm = lcm of numerator numbers/hcf of denominator numbers.

**Numerator numbers are 5, 3, and 9

Their l.c.m = 45.

**denominator numbers are 8, 4 and 20

Their hcf = 4

Lcm = 45/4 or 11.25

2. Find the hcf of 0.8, 2/9, 0.72 and 8/9

__Solution__

First change all the decimal numbers to fractions.

0.8 = 8/10 = 4/5 when reduced.

0.72 = 72/100 = 18/25.

Our fractions will be

4/5, 2/9, 18/25 and 8/9.

h.c.f = hcf of numerator numbers/lcm of denominator numbers

= hcf of 4, 2, 18 and 8 / lcm of 5, 9, 25 and 9

= 2/225.

**EXERCISE**

Find the lcm and hcf of the following numbers

1. 20/27 and 4/9

2. 225 and 0.6

3. 18/25, 21/40 and 12/25

4. 16/25, 0.4 and 12/15.

**Other problems**

1. If two alarm clocks are set such that they ring every 12 minutes and 20 minutes. After how many minutes will they ring together?

**Solution**

The lcm of 12 and 20 will give us when they will ring together.

2 |
12 |
20 |

2 |
6 |
10 |

3 |
3 |
5 |

5 |
1 |
5 |

1 |
1 |

Lcm = 2 x 2 x 3 x 5

= 60

Therefore, they will ring together after 60 minutes.

2. Smart, Joy and Ken usually visit Mama T's cafe every 25 minutes, 30 minutes and 50 minutes respectively. If the three friends met at the cafe at 11:20 am, what time will they meet again?

**Solution**

Find the lcm of 25, 30 and 50

Lcm = 150

That means they will meet every 150 minutes, to get the required time, convert the 150 minutes to hours

= 150/60

= 2 hours 30 minutes.

Add 2 hours 30 mins to 11:20 am

= 13:50

Which is 01:50 pm.

3. The l.c.m of 12 and another number is 60. If their h.c.f is 4, find the number.

**Solution**

**Product of the two numbers = product of their l.c.m and h.c.f**

i.e. if the two numbers are **a** and **b, **then

**a x b = l.c.m x h.c.f**

in the above question, a = 12, b = ?, l.c.m = 60, h.c.f = 4.

12 x b = 60 x 4

12b = 240

b = 240/12

b = 20.

** Practice questions**

1. After a budget planning consultation Mr. A, B and C decided that they will change their car after every 16 months, 20 months and 30 months respectively. If they all bought their cars together in January 10, 2017, when next will the buy together?

2. The hcf of 18 and another number is 6, if their l.c.m is 72, find the number.

3. The lcm of 6 and another number is 12. if their h.c.f is 2, find the number.

4. Three vigilante men usually blow their whistles every 20, 15 and 25 minutes. If they blow together at 10:00pm, when next will they blow together?

**PRE:** lcm of algebraic expression