What would be the maximum value of Q in the equation 5P9 + 3R7 + 2Q8 = 1114?
9
Solving the Digit Equation: Finding the Maximum Value of Q
The problem asks for the maximum possible value of the digit Q in the addition equation:
\(5P9 + 3R7 + 2Q8 = 1114\)
Here, P, R, and Q represent single digits (from 0 to 9). We can analyze this addition by setting it up vertically, like a standard addition problem:
| Hundreds | Tens | Units | |
|---|---|---|---|
| 5 | P | 9 | |
| 3 | R | 7 | |
| + | 2 | Q | 8 |
| 11 | 1 | 4 | |
Let's analyze the addition column by column, starting from the rightmost (units) column.
Units Column Analysis
The units column involves the digits 9, 7, and 8. Their sum is:
\(9 + 7 + 8 = 24\)
In the result 1114, the units digit is 4. This matches the units digit of 24. The '2' from 24 is a carry-over to the tens column.
Tens Column Analysis
The tens column involves the digits P, R, and Q, plus the carry-over 2 from the units column. The sum of the tens column must result in a number whose units digit is 1 (as seen in 1114), with a carry-over to the hundreds column.
The sum in the tens column is \(P + R + Q + 2\). The units digit of this sum is 1, and there is a carry-over to the hundreds column. The only single-digit sums that result in a units digit of 1 with a carry-over are 11 or 21. Since P, R, and Q are single digits (maximum value 9), their maximum sum is \(9+9+9=27\). Adding the carry-over 2, the maximum possible sum for \(P+R+Q+2\) is \(27+2=29\). So, the sum \(P+R+Q+2\) must be 11 or 21.
- If \(P + R + Q + 2 = 21\), then \(P + R + Q = 19\). The maximum possible sum for P, R, and Q is \(9+9+9=27\), so \(P+R+Q=19\) is possible. In this case, the carry-over to the hundreds column would be 2.
- If \(P + R + Q + 2 = 11\), then \(P + R + Q = 9\). The maximum possible sum for P, R, and Q is 27, so \(P+R+Q=9\) is possible. In this case, the carry-over to the hundreds column would be 1.
Hundreds Column Analysis
The hundreds column involves the digits 5, 3, and 2, plus the carry-over from the tens column. The sum must result in the hundreds digit 1 and thousands digit 1 (forming 11 in 1114).
The sum of the hundreds digits is \(5 + 3 + 2 = 10\).
Now, let's consider the carry-over from the tens column:
- If the carry-over from the tens column was 2, the sum in the hundreds column would be \(10 + 2 = 12\). This would mean the result would have 2 in the hundreds place and 1 carried over to the thousands place, resulting in a number like 12xx. This does not match the sum 1114.
- If the carry-over from the tens column was 1, the sum in the hundreds column would be \(10 + 1 = 11\). This would mean the result has 1 in the hundreds place and 1 carried over to the thousands place, resulting in 11xx. This matches the sum 1114.
Therefore, the carry-over from the tens column must be 1. This confirms that the sum in the tens column was 11, leading to the equation:
\(P + R + Q + 2 = 11\)
Subtracting 2 from both sides, we get:
\(P + R + Q = 9\)
Finding the Maximum Value of Q
We have the equation \(P + R + Q = 9\), where P, R, and Q are digits from 0 to 9. To find the maximum possible value of Q, we need to assign the smallest possible values to P and R.
The smallest possible value for a digit (P or R) is 0.
Let's set \(P = 0\) and \(R = 0\). Substituting these values into the equation:
\(0 + 0 + Q = 9\)
\(Q = 9\)
The value Q = 9 is a valid digit (it is between 0 and 9).
Verification
Let's check if \(P=0\), \(R=0\), and \(Q=9\) satisfy the original equation:
- 5P9 becomes 509
- 3R7 becomes 307
- 2Q8 becomes 298
Adding these numbers:
\(509 + 307 + 298\)
\(509 + 307 = 816\)
\(816 + 298 = 1114\)
The sum is indeed 1114. This confirms that Q can be 9.
Since setting P and R to their minimum possible values (0) gives Q = 9, and 9 is a valid digit, the maximum value of Q is 9.
Revision Table: Equation Analysis
| Column | Digits | Sum | Result Digit | Carry-over | Equation |
|---|---|---|---|---|---|
| Units | 9, 7, 8 | \(9+7+8=24\) | 4 | 2 | - |
| Tens | P, R, Q + carry 2 | \(P+R+Q+2\) | 1 | 1 | \(P+R+Q+2=11\) |
| Hundreds | 5, 3, 2 + carry 1 | \(5+3+2+1=11\) | 11 | - | Matches 11 in 1114 |
From the tens column analysis, we derived the key relationship: \(P + R + Q = 9\).
Additional Information: Digit Puzzles
Problems like this, where letters represent digits in a mathematical equation, are often called alphametic or cryptarithmetic puzzles. The goal is usually to find the digit that each letter represents.
Key rules and tips for solving digit puzzles:
- Each letter represents a unique digit (0-9), unless stated otherwise (in this problem, P, R, and Q are just digits, not necessarily unique or different from 5, 3, 2, 9, 7, 8). In this specific question, P, R, and Q must be digits from 0-9.
- The first digit of a number (like the 5 in 5P9, 3 in 3R7, 2 in 2Q8) cannot be 0. However, P, R, or Q can be 0 if they are not the leading digit.
- Analyze the columns starting from the rightmost column (units place).
- Pay close attention to carry-overs between columns.
- Look for constraints based on maximum/minimum possible sums of digits.
- Use logical deduction and trial-and-error to find possible digit values.
In our problem, P, R, and Q were digits from 0 to 9. The maximum value for Q is achieved when P and R are minimized, which is 0.
Similar Questions
What is the solution of the equation x log 10 (10/3) + log 10 3 = log 10 (2 + 3 x) + x?
If a 3= 335 + b 3and a = 5 + b, then what is the value of a + b (given that a > 0 and b > 0)?
What is the value of u in the system of equations 3 (2u + v) = 7uv, 3 (u + 3v) = 11uv?
Five years ago, Ram was three times as old as Shyam. Four years from now, Ram will be only twice as old as Shyam. What is the present age of Ram?
The values of x and y from the equations x - y = 6 and x/3 + y/2 = 12 are:
What is the value of x, 2x/3 + y/ 2 = 4 and x/3 - y/2 = 1?
How many integral values of x and y satisfy the equation 5x + 9y = 7, where -500 < x < 500 and -500 < y < 500?
A person carries Rs. 500 and wants to buy apples and oranges out of it. If the cost of one apple is Rs. 5 and the cost of one orange is Rs. 7 then what is the number of ways in which a person can buy both apples and oranges using total amount?
How many pairs of natural numbers are there such that the difference of their squares is 35?
What is the maximum value of the expression \(\frac{1}{{{x^2}\; + \;5x\; + \;10}}?\)
Important Questions from Linear Equation in 2 Variable
If 2 x + 3 y = 17;
2 x+2 - 3 y+1 = 5
then the values of x and y are:
The solution of pair of linear equations \(\dfrac{1}{2}x+\dfrac{2}{3}y=-1,x-\dfrac{1}{3}y=3\) by the elimination method, is:
Kumar tried his skill at shooting at a fun fair. He has to hit the target and if he hits the target he gets 1 Rs. and if he misses he has to pay 50 paise. He attempted 25 shots and won 10 Rs. In how many did he hit the target?
Shyam spent half of his money and was left with as many as he had rupees before, but with half as many rupees as he had paise before. Which of the following is a possible amount of money he is left with?
Two bus tickets from city A to B and three tickets from city A to C cost Rs. 77, but three tickets from city A to B and two tickets from city A to C cost Rs. 73. What are the fares for cities B and C from A?
