Question

In a certain game, each player scores either 2 points or 5 points. If \(n\) players score 2 points and \(m\) players score 5 points, and the total number of points scored is \(50,\) what is the least possible positive difference between \(n\) and \(m ?\)

Short answer

The least possible positive difference is 3.

Step-by-step solution

Set Up the Equation

The total points scored can be found using the equation: \[ 2n + 5m = 50 \] where \(n\) is the number of players scoring 2 points, and \(m\) is the number of players scoring 5 points.

Determine the Range of Possible Values

Since both \(n\) and \(m\) are non-negative integers, we need to find integer solutions for \(2n + 5m = 50\).

Find Integer Solutions for m

Solve for integer values of \(m\): Rearrange the equation to isolate \(n\): \[ n = \frac{50 - 5m}{2} \] For \(n\) to be an integer, \(50 - 5m\) must be even. Test different values for \(m\) to find suitable pairs \((n, m)\).

Test Possible Values of m

Check different integer values of \(m\) to see when \(n\) is also an integer: For example: - If \(m = 0\): \[ n = \frac{50 - 0}{2} = 25 \] - If \(m = 2\): \[ n = \frac{50 - 10}{2} = 20 \] - If \(m = 4\): \[ n = \frac{50 - 20}{2} = 15 \] - If \(m = 6\): \[ n = \frac{50 - 30}{2} = 10 \] - If \(m = 8\): \[ n = \frac{50 - 40}{2} = 5 \] - If \(m = 10\): \[ n = \frac{50 - 50}{2} = 0 \]

Calculate Differences

Find the differences between \(n\) and \(m\) for each pair: - For \((25, 0)\) difference is \(25 - 0 = 25\) - For \((20, 2)\) difference is \(20 - 2 = 18\) - For \((15, 4)\) difference is \(15 - 4 = 11\) - For \((10, 6)\) difference is \(10 - 6 = 4\) - For \((5, 8)\) difference is \(5 - 8 = 3\) - For \((0, 10)\) difference is \(0 - 10 = 10\)

Find the Least Possible Positive Difference

Identify the smallest positive difference from the calculations: The smallest positive difference is \(3\) for the pair \( (5, 8) \).

Key concepts

headline of the respective core concept

Many math problems involve finding integers that satisfy an equation. Integers are whole numbers, which can be positive, negative, or zero. In our problem, we need to solve for non-negative integers that fit the equation: \[2n + 5m = 50\] This tells us how points by players are scored in such a way that both players' scores combined equal 50. Our task is to find values of \(n\) (number of players scoring 2 points) and \(m\) (number of players scoring 5 points), ensuring both are integers. For this, we try out several values for \(m\) and solve for \(n\) until we find valid pairs.

headline of the respective core concept

Linear equations are equations where the variables appear to the first degree, i.e., they are not squared or cubed, etc. Our given equation is a linear equation: \[2n + 5m = 50\] We rearrange it to make one variable the subject. For example, to isolate \(n\), we rearrange as: \[ n = \frac{50 - 5m}{2} \] This form allows us to test different values of \(m\) and determine corresponding values of \(n\). The goal is to ensure that both \(n\) and \(m\) remain integers for feasible solutions.

headline of the respective core concept

Solving math problems often involves breaking them down step by step. Let’s see the steps we took:1. Set up the equation reflecting total points scored: \[2n + 5m = 50\]2. Rearrange the equation to solve for \(n\): \[n = \frac{50 - 5m}{2}\]3. Test different values for \(m\) to find integer solutions. Check for \(m = 0, 2, 4, 6, 8, 10\).4. Calculate \(n\) with these values of \(m\) and ensure \(n\) also remains an integer: - For \(m = 0 : n = 25\) - For \(m = 2 : n = 20\) - For \(m = 4 : n = 15\) - and so forth5. Compute differences between \(n\) and \(m\) for each pair and identify the smallest positive difference.

headline of the respective core concept

Mathematical reasoning is all about using steps and logic to solve problems. In our problem, we used logical reasoning to find our solution in several stages:- First, we recognized that the total points scored could be expressed as a linear equation.- Next, we realized integer values were necessary to make sense in the context of the problem.- Logical reasoning then led us to test specific values of \(m\) to find corresponding integer solutions of \(n\).Finally, evaluating which pair gave the smallest positive difference involved a bit of comparison and analytical thinking. This is how mathematical reasoning helps us solve puzzles systematically.