Problem 35 - Fundamentals of Mathematics

Question

\(3 \frac{1}{5}\) of what number is \(1 ?\)

Step-by-Step Solution

Answer

The number is \(\frac{5}{16}\).

Step 1: Understand the Problem

We need to find a number such that when it is multiplied by \(3 \frac{1}{5}\), the result is 1. This setup can be expressed as an algebraic equation.

Step 2: Convert Mixed Number to Improper Fraction

Convert \(3 \frac{1}{5}\) into an improper fraction. To do that, multiply the whole number 3 by the denominator 5, and add the numerator 1. So, \(3 \frac{1}{5} = \frac{16}{5}\).

Step 3: Express the Problem as an Equation

Let the number we’re looking for be \(x\). The equation based on the problem statement is \(\frac{16}{5} \times x = 1\).

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation by \(\frac{16}{5}\). This can be done by multiplying by the reciprocal. So, \(x = 1 \times \frac{5}{16}\).

Step 5: Simplify the Expression

Perform the multiplication: \(x = \frac{5}{16}\). This is the number we are looking for.

Key Concepts

Algebraic Equations

Algebraic equations are the foundation of many mathematical problems and concepts. They are like a balance scale, where both sides need to equal each other. In our exercise, we needed to find a number that, when multiplied by a mixed number (\(3 \frac{1}{5}\)), equals 1. This type of problem can be neatly organized and solved using an algebraic equation.
To express this situation, we let the number be \(x\). We then set up the equation \(\frac{16}{5} \times x = 1\). Here, \(\frac{16}{5}\) is derived from converting the mixed number to an improper fraction (more on that later).
The goal is to solve for \(x\), meaning to find the numerical value which satisfies the equation. Solving typically involves performing operations to isolate \(x\) on one side of the equation. In our problem, multiplication by the reciprocal of \(\frac{16}{5}\) simplifies both sides and leaves \(x\) by itself, giving us the solution: \(x = \frac{5}{16}\).

• Identify the equation based on the problem statement.
• Use mathematical operations to isolate the unknown variable.
• Simplify where possible to find the solution.

Mixed Numbers

Mixed numbers combine a whole number with a fraction. They are often seen in everyday life and can make expressions more relatable. In equations, however, it's often easier to work with improper fractions instead.
A mixed number like \(3 \frac{1}{5}\) is essentially the sum of 3 (a whole number) and \(\frac{1}{5}\) (a fraction). While mixed numbers provide clarity in certain contextual scenarios, mathematics often requires conversion for ease of calculation.
The conversion to an improper fraction involves the following steps:

• Multiply the whole number by the denominator of the fraction part, which in our case is \(3 \times 5 = 15\).
• Add the numerator of the fraction to this result, thus \(15 + 1 = 16\).
• Write the new numerator over the original denominator, so we get \(\frac{16}{5}\).

This improper fraction makes it easier to perform multiplication or division in algebraic equations.

Improper Fractions

Improper fractions have a numerator that is larger than or equal to the denominator. While they can seem awkward at first, they are particularly useful in algebra and higher-level math for straightforward calculations.
An improper fraction helps avoid the complication of handling both a whole number and a fractional part, as is the case with mixed numbers.
In our example, converting \(3 \frac{1}{5}\) to \(\frac{16}{5}\) allowed us to easily perform operations needed in the algebraic equation we formed.
Here’s how improper fractions serve us better:

• They streamline calculations, letting you perform multiplication and division directly without separate steps for the whole and fractional parts.
• In algebra, they easily fit into formulas and equations without additional conversion, saving time and reducing error.
• Expressing numbers as improper fractions can reveal patterns or ratios more clearly.

Understanding how to shift between mixed numbers and improper fractions is crucial for solving problems accurately and efficiently.