(a) The product of the digits of a 2-digit number is 18. When 27 is subtracted from the number, the digits interchange their places. Find the number.
OR
(b) Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers.
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Step-by-Step Solution
Step 1: Set up equations for problem (a)
Let the two-digit number be represented as 10x + y, where x is the tens digit and y is the units digit. From the problem statement, the product of the digits is 18, which gives us the equation xy=18. When 27 is subtracted from the number, the digits interchange their places, meaning the new number is 10y + x. This gives us the second equation (10x+y)−27=10y+x.
Step 2: Solve the system of equations for problem (a)
Simplify the second equation: 10x+y−27=10y+x⟹9x−9y=27⟹x−y=3. From this, we get x=y+3. Substitute this into the first equation: (y+3)y=18, which simplifies to a quadratic equation y2+3y−18=0. Factoring this quadratic equation gives (y+6)(y−3)=0. Since y is a digit, it must be positive, so y=3. Substituting y=3 back into x=y+3 gives x=6.
Step 3: Determine the number for problem (a)
With x=6 and y=3, the two-digit number is 10(6)+3=63.
Step 4: Set up equations for problem (b)
Let the two numbers be a and b. Their initial ratio is 5:6, so ba=65, which simplifies to 6a=5b. When 8 is subtracted from each number, the new ratio is 4:5, so b−8a−8=54. This cross-multiplies to 5(a−8)=4(b−8).
Step 5: Solve the system of equations for problem (b)
Expand the second equation: 5a−40=4b−32, which simplifies to 5a−4b=8. From the first equation, 6a=5b, we can express b as b=56a. Substitute this expression for b into the simplified second equation: 5a−4(56a)=8. Multiply by 5 to clear the denominator: 25a−24a=40, which gives a=40. Substitute a=40 back into b=56a to find b=56(40)=48.
Step 6: Determine the numbers for problem (b)
The two numbers are a=40 and b=48.