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The ratio of two numbers is 7:5, and their H.C.F. is 6. What is their L.C.M.?
- a)148
- b)156
- c)210
- d)270
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The ratio of two numbers is 7:5, and their H.C.F. is 6. What is their...
Let the numbers be 7x and 5x. As 7 and 5 are co-prime pairs.
∴ Their H.C.F. = x = 6 (given).
=> The numbers are 42 and 30.
L.C.M. of 42 and 30 = 210.
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The ratio of two numbers is 7:5, and their H.C.F. is 6. What is their...
Understanding the Ratio and H.C.F.
The ratio of two numbers, say \( a \) and \( b \), is given as 7:5. This means we can express \( a \) and \( b \) in terms of a common multiplier \( k \):
- \( a = 7k \)
- \( b = 5k \)
We are also informed that their Highest Common Factor (H.C.F.) is 6.
Finding the Values of the Numbers
Since the H.C.F. of \( a \) and \( b \) is 6, we can set \( 5k \) and \( 7k \) equal to the H.C.F. multiplied by a factor. This means:
- H.C.F. = 6 = \( 5k \) / 5 or \( 7k \) / 7
Thus, we can express \( k \):
- \( k = 6 \)
Now substituting \( k \) back to find \( a \) and \( b \):
- \( a = 7k = 7 \times 6 = 42 \)
- \( b = 5k = 5 \times 6 = 30 \)
Calculating L.C.M.
The formula for calculating the Least Common Multiple (L.C.M.) using H.C.F. is:
- \( \text{L.C.M.} \times \text{H.C.F.} = a \times b \)
Substituting the known values:
- \( a \times b = 42 \times 30 \)
Calculating \( a \times b \):
- \( 42 \times 30 = 1260 \)
Using the relationship between L.C.M. and H.C.F.:
- \( \text{L.C.M.} \times 6 = 1260 \)
Solving for L.C.M.:
- \( \text{L.C.M.} = 1260 / 6 = 210 \)
Conclusion
Thus, the L.C.M. of the two numbers is:
- 210 (option 'C')
The ratio of two numbers, say \( a \) and \( b \), is given as 7:5. This means we can express \( a \) and \( b \) in terms of a common multiplier \( k \):
- \( a = 7k \)
- \( b = 5k \)
We are also informed that their Highest Common Factor (H.C.F.) is 6.
Finding the Values of the Numbers
Since the H.C.F. of \( a \) and \( b \) is 6, we can set \( 5k \) and \( 7k \) equal to the H.C.F. multiplied by a factor. This means:
- H.C.F. = 6 = \( 5k \) / 5 or \( 7k \) / 7
Thus, we can express \( k \):
- \( k = 6 \)
Now substituting \( k \) back to find \( a \) and \( b \):
- \( a = 7k = 7 \times 6 = 42 \)
- \( b = 5k = 5 \times 6 = 30 \)
Calculating L.C.M.
The formula for calculating the Least Common Multiple (L.C.M.) using H.C.F. is:
- \( \text{L.C.M.} \times \text{H.C.F.} = a \times b \)
Substituting the known values:
- \( a \times b = 42 \times 30 \)
Calculating \( a \times b \):
- \( 42 \times 30 = 1260 \)
Using the relationship between L.C.M. and H.C.F.:
- \( \text{L.C.M.} \times 6 = 1260 \)
Solving for L.C.M.:
- \( \text{L.C.M.} = 1260 / 6 = 210 \)
Conclusion
Thus, the L.C.M. of the two numbers is:
- 210 (option 'C')
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