Q5.Here Are Two Fractions.7/5 And 5/7Work Out Which Of The Fractions Is Closer To 1You Must Show All

Q5.Here Are Two Fractions.7/5 And 5/7Work Out Which Of The Fractions Is Closer To 1You Must Show All

Understanding how to compare fractions and determine which is closer to a specific number, such as 1, is a fundamental skill in mathematics. When faced with the fractions 7/5 and 5/7, it might seem challenging at first glance to see which one is nearer to 1. However, by systematically analyzing these fractions, converting them into decimal form, or using other mathematical techniques, we can confidently identify the fraction closest to 1. This guide will explore various methods to compare these fractions thoroughly, ensuring a clear understanding of the process.

Understanding Fractions and Their Relationship to 1

What Does It Mean for a Fraction to Be Close to 1?

A fraction is close to 1 when its value is nearly equal to 1. Since 1 can be written as a fraction itself (1/1), comparing fractions to 1 involves looking at how their values differ from 1. For example:

• Fractions greater than 1 are called improper fractions, such as 7/5.
• Fractions less than 1 are called proper fractions, such as 5/7.

The goal is to see which of these two fractions—7/5 or 5/7—is numerically closest to 1.

Why Is This Important?

Understanding how to compare fractions to 1 is essential in many areas of mathematics, including ratios, proportions, and real-world measurements. It helps develop intuition about the size of fractions and their relative position to whole numbers.

Method 1: Converting Fractions to Decimals

One of the simplest ways to compare fractions is to convert them into decimal form. Decimals are often easier to compare visually or through simple subtraction.

Step-by-Step Conversion

1. Convert 7/5 into a decimal:

• Divide 7 by 5: 7 ÷ 5 = 1.4

2. Convert 5/7 into a decimal:

• Divide 5 by 7: 5 ÷ 7 ≈ 0.7142857

Comparing the Decimals to 1

• 7/5 ≈ 1.4
• 5/7 ≈ 0.714

Now, compare how close each decimal is to 1:

• Distance of 7/5 from 1: |1.4 - 1| = 0.4
• Distance of 5/7 from 1: |0.7142857 - 1| ≈ 0.2857

Since 0.2857 is less than 0.4, 5/7 is closer to 1 than 7/5.

Conclusion from Method 1:
The fraction 5/7 is closer to 1 than 7/5.

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Method 2: Cross-Multiplication Technique

Another effective method for comparing two fractions without converting to decimals involves cross-multiplication.

How Cross-Multiplication Works

Given two fractions, a/b and c/d:

• Calculate a × d and c × b.
• The fraction with the smaller product is closer to 1 if we are comparing their proximity to 1.

Applying Cross-Multiplication to Our Fractions

Compare 7/5 and 5/7:

• Cross-multiplied products:
• 7 × 7 = 49
• 5 × 5 = 25

But, we need to compare their difference from 1, so a more precise approach is to compare their absolute difference from 1 by considering their distance in terms of numerator and denominator.

Alternatively, we can analyze their deviation from 1 by rewriting each fraction:

• For 7/5:


• 7/5 = 1 + 2/5


• For 5/7:


• 5/7 ≈ 0.7143, which is less than 1, so:


• 1 - 5/7 ≈ 0.2857

From this, it's clear that 5/7 is less than 1, but its difference from 1 is approximately 0.2857, while 7/5 exceeds 1 by 0.4, as previously calculated.

Conclusion from Method 2:
Again, the smaller difference indicates 5/7 is closer to 1.

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Method 3: Visual Representation and Intuitive Understanding

Visual aids can help deepen understanding, especially for learners who find number lines helpful.

Plotting on a Number Line

• Mark 1 on the number line.
• Plot 7/5:
• Since 7/5 = 1.4, place a point slightly to the right of 1.
• Plot 5/7:
• Since 5/7 ≈ 0.714, place a point slightly to the left of 1.

The visual distance from these points to 1 confirms that 5/7 is closer.

Implication of Visuals

Visual comparison reinforces the numeric calculations, solidifying the conclusion that 5/7 is the fraction nearer to 1.

Summary: Which Fraction Is Closer to 1?

By analyzing the fractions through multiple methods:

• Decimal conversion shows 5/7 is approximately 0.714, closer to 1 than 7/5, which is 1.4.
• Cross-multiplication and difference calculations support this conclusion.
• Visual representation confirms that 5/7 is the closer fraction.

Final Answer: The fraction 5/7 is closer to 1 than 7/5.

Additional Tips for Comparing Fractions to 1

• Always consider converting to decimals for quick comparisons.
• Use cross-multiplication to avoid decimal inaccuracies.
• Visual tools like number lines can aid understanding.
• Remember, fractions less than 1 are closer to 1 if their numerator is just slightly less than the denominator, and vice versa for fractions greater than 1.

Practice Problems to Reinforce Your Skills

1. Compare 9/8 and 8/9 and determine which is closer to 1.
2. Is 11/10 or 10/11 nearer to 1? Show your reasoning.
3. Find which of the following fractions is closest to 1: 13/12, 12/13, or 14/13.

Practicing these types of problems enhances your ability to quickly compare fractions and develop mathematical intuition.

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In conclusion, understanding and comparing fractions relative to 1 involves multiple strategies, including decimal conversion, cross-multiplication, and visual aids. Applying these methods to the fractions 7/5 and 5/7 demonstrates that 5/7 is nearer to 1, a conclusion supported by consistent calculations and visual confirmation. Mastery of these techniques is a valuable component of mathematical literacy, enabling learners to analyze and interpret fractions confidently in diverse contexts.