a certain number of two digits is equal to five times the sum of digits. three times the larger digit exceeds three times the smaller by three. find the number
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a certain number of two digits is equal to five times the sum of digits. three times the larger digit exceeds three times the smaller by three. find the number
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Verified answer
Answer:
twice the product of its digits. Find the number.
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Solution
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Let the number be 10x+y
According to question, we have,
10x+y=5(x+y)
=>10x+y=5x+5y
=>5x=4y
Also, 10x+y−2xy=5
=>10x+
4
5x
−2x×
4
5x
=5
=>10x+
4
5x
−
4
10x
2
=5
=>40x+5x−10x
2
=20
=>10x
2
−45x+20=0
=>2x
2
−9x+4=0
=>2x
2
−8x−x+4=0
=>2x(x−4)−1(x−4)=0
=>(2x−1)(x−4)=0
=>x=4
y=
4
5x
=
4
5×4
=5
Required number is 10×4+5
=45
Answer:
x== the larger digit
y= the smaller digit
"Three times the larger digit exceeds three times the smaller by three" translates as
3x=3y%2B3<-->3%2Ax=3%28y%2B1%29<-->x=y%2B1 .
That narrows the possibilities to
10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, and 98.
What other clue do we have?
t= the tens digit
u= the ones digit
t%2Bu= the sum of the digits
10t%2Bu= the value of the number
The "number of two digits is equal to five times the sum of the digits"
translates into the equation
10t%2Bu=5%28t%2Bu%29<-->10t%2Bu=5t%2B5u<-->10t-5t=5u-u<-->5t=4u<-->t=%284%2F5%29%2Au .
Maybe by now you realize that the number is highlight%2845%29.
Otherwise, t=%284%2F5%29%2Au tells you that t is the smaller digit, so u=t%2B1 .
system%28u=t%2B1%2C5t=4u%29--->system%28u=t%2B1%2C5t=4%28t%2B1%29%29--->system%28u=t%2B1%2C5t=4t%2B4%29--->system%28u=t%2B1%2C5t-4t=4%29--->system%28u=t%2B1%2Ct=4%29--->system%28u=4%2B1%2Ct=4%29--->highlight%28system%28u=5%2Ct=4%29%29