Question

Toni had a backlog of work to do. On Monday she did \(\frac{1}{4}\) of the backlog. If on Tuesday she does the same amount of work (e.g., processes the exact same number of files), what part of the backlog will be left? A. \(\frac{1}{8}\) B. \(\frac{1}{4}\) C. \(\frac{1}{2}\) D. \(\frac{2}{3}\)

Short answer

The remaining backlog is \(\frac{1}{2}\), which is option C.

Step-by-step solution

Determine the fraction of backlog completed each day

Toni did \(\frac{1}{4}\) of the backlog on Monday. Since it's given that on Tuesday she does the same amount of work, she also did \(\frac{1}{4}\) of the backlog on Tuesday.

Calculate the total fraction of the backlog completed

Add the fractions of the backlog completed on both days: \(\frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\). Therefore, Toni completed \(\frac{1}{2}\) of the backlog.

Determine the remaining backlog

Subtract the completed fraction from 1 (the whole backlog): \(\text{Remaining backlog} = 1 - \frac{1}{2} = \frac{1}{2}\).

Match the result with the given options

The fraction of the backlog remaining is \(\frac{1}{2}\), which corresponds to option C.

Key concepts

Fraction Addition

Fraction addition is a fundamental concept in arithmetic that involves summing fractions to get a total amount. In our example, Toni did \(\frac{1}{4}\) of her work on Monday and another \(\frac{1}{4}\) on Tuesday. To find out the total part of the backlog completed, we add these fractions.

Fractions must have a common denominator to be added. Here, both fractions already have the same denominator (4), making our task simple. We just add the numerators: \(\frac{1}{4} + \frac{1}{4} = \frac{2}{4}\).

We can then simplify \(\frac{2}{4}\) to \(\frac{1}{2}\). So, Toni completed half of her backlog. Understanding how to add fractions with common denominators helps in solving many real-life problems involving parts and wholes.

Basic Algebra

Basic algebra involves understanding and manipulating mathematical symbols to solve problems. In this exercise, we used simple algebraic steps to find out how much backlog remained.

After figuring out that Toni did \(\frac{1}{2}\) of her work, we needed to find the remaining part. Since a whole is represented by 1, we subtract the part completed from the whole: \1 - \frac{1}{2} = \frac{1}{2}\.

Algebra helps us transition from adding fractions to subtracting them to find what's left. It's important to remember that subtraction here follows the principle of taking something away from a total set. This exercise shows how basic algebra can be used alongside fraction addition to solve more comprehensive problems.

Problem-Solving Skills

Problem-solving skills are crucial in mathematics and are developed through practice and understanding of concepts. Let’s review how these skills apply to our example with Toni.

Step-by-step problem solving involves:
1. Identifying the problem: What fraction of the backlog remains?
2. Breaking down the problem: Determine what part was completed each day.
3. Applying known concepts: Add the fractions and use basic algebra for subtraction.
4. Concluding: Interpret the result and match it with given options.

In this case, identifying that the nature of the problem involves fractions and a straightforward subtraction was crucial. Practicing problems like these helps build a systematic approach to solving different math problems, enhancing both accuracy and efficiency.