Problem 25 - A History of Mathematics: An Introduction

Question

Solve Epigram 145: A. Give me ten coins and I have three times as many as you. B. And if I get the same from you, I have five times as much as you? How many coins does each have?

Step-by-Step Solution

Answer

Answer: A has 17 coins and B has 19 coins.

Step 1: Rewrite the statements as equations

We can rewrite the two pieces of information as equations: 1. (a + 10) = 3 * (b - 10) 2. (b + 10) = 5 * (a - 10)

Step 2: Solve for one variable in terms of the other

Now let's solve the first equation for 'a' in terms of 'b': a + 10 = 3 * (b - 10) a = 3b - 30 - 10 a = 3b - 40

Step 3: Substitute the expression for 'a' into the second equation

Now, let's substitute the expression for 'a' from step 2 into the second equation: (b + 10) = 5 * (3b - 40 - 10) b + 10 = 15b - 250

Step 4: Solve for 'b'

Now, let's solve the equation from step 3 for 'b': 14b = 260 b = \frac{260}{14} b = 18.57 Since the number of coins has to be a whole number, we need to round the result to the nearest whole number. b = 19

Step 5: Solve for 'a'

Let's substitute the value of 'b' (19) back into the equation for 'a' from step 2 to find the value of 'a': a = 3 * 19 - 40 a = 57 - 40 a = 17

Step 6: Interpret the results

Finally, we can conclude that A has 17 coins and B has 19 coins.

Key Concepts

Algebraic Equations

In this problem, algebraic equations are used to transform word problems into mathematical expressions that can be solved systematically. An algebraic equation is a mathematical statement indicating the equality of two expressions. In our exercise, two statements were converted into equations:

• (a + 10) = 3 \( \times \) (b - 10)
• (b + 10) = 5 \( \times \) (a - 10)

These expressions capture the relationship between the coins that individuals A and B possess. The variable \( a \) represents the number of coins A has before receiving coins, and \( b \) represents the number B has initially.
To solve these equations, we first express one variable in terms of another (this is called substitution). This approach simplifies the problem, allowing us to manage one equation at a time. Understanding how to rewrite a problem in algebraic terms is a core skill in algebra that helps in solving complex and real-life problems more efficiently.

Logical Reasoning in Mathematics

Logical reasoning plays a crucial role in mathematical problem-solving. In this exercise, logical reasoning helps in the translation of verbal statements into mathematical expressions.
The logic begins with deducing what needs to be done from the given information. From statement A's perspective, gaining ten coins led to tripling the coins compared to B. Similarly, from statement B's perspective, adding ten coins led to having five times what A possessed.

• First, identify the relationships and constraints given.
• Convert these relationships into equations.
• Systematically solve these equations to reach a conclusion.

By applying a step-by-step logical thought process, one can deconstruct seemingly complex problems into manageable algebraic equations, as demonstrated in this exercise. Logical reasoning is thus the bridge that guides the transition of verbal narratives to mathematic solutions.

Number Theory

Number theory, a branch of pure mathematics, often involves understanding the properties and behaviors of numbers. In our problem, number theory comes into play when determining the integer values that make the solutions meaningful.
The concept is especially important because while solving algebraic equations, the calculated solutions must adhere to the context of the problem—in this case, the whole number of coins.

• After solving an equation like \( b = \frac{260}{14} \), the result should ideally be an integer since coins cannot be fractional.
• We find \( b \) to be approximately 18.57, which requires rounding to the nearest whole number for a practical solution.

It's essential to validate that the final numbers satisfy the original conditions of the problem. This exercise is a classic application of number theory, ensuring the plausibility and correctness of each solution in the problem-solving process.