How to Simplify 75/90 to Its Lowest Terms

The fraction 75/90, when simplified to its simplest form (also known as lowest terms), is 5/6.

To reach this result, the numerator (75) and the denominator (90) are both divided by their greatest common factor, which is 15. In mathematical terms: 75 ÷ 15 = 5 90 ÷ 15 = 6

While the answer is straightforward, understanding the process of fraction reduction is a fundamental skill in mathematics. This involves exploring various methods of simplification, converting the value into different formats such as decimals and percentages, and recognizing the practical utility of these numbers in everyday life.

The Mathematical Structure of 75/90

Before diving into the mechanics of simplification, it is essential to understand what the fraction 75/90 represents. In any fraction, we have two primary components:

1. The Numerator (75): This represents the "part" or the number of units we are considering.
2. The Denominator (90): This represents the "whole" or the total number of equal parts that make up a single entity.

When we say 75/90, we are describing a scenario where 75 parts are taken out of a total of 90. This could be 75 correct answers out of 90 questions on an exam, or 75 cents out of a 90-cent total.

Simplification does not change the value of the fraction. Instead, it expresses that same value using the smallest possible whole numbers. This makes the number easier to visualize, communicate, and use in further calculations.

Method 1: The Greatest Common Factor (GCF) Approach

The most direct way to simplify 75/90 is to find the Greatest Common Factor (GCF). The GCF is the largest positive integer that divides both the numerator and the denominator without leaving a remainder.

Step 1: List the Factors of 75

To find the factors of 75, we identify all whole numbers that can be multiplied together to equal 75:

• 1 × 75 = 75
• 3 × 25 = 75
• 5 × 15 = 75

The factors of 75 are: 1, 3, 5, 15, 25, 75.

Step 2: List the Factors of 90

Next, we do the same for the denominator, 90:

• 1 × 90 = 90
• 2 × 45 = 90
• 3 × 30 = 90
• 5 × 18 = 90
• 6 × 15 = 90
• 9 × 10 = 90

The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

Step 3: Identify the Greatest Common Factor

Now, we compare the two lists to find the largest number present in both:

• Common Factors: 1, 3, 5, 15.
• Greatest Common Factor: 15.

Step 4: Divide Both Numbers

To reduce the fraction, divide both the top and the bottom by 15:

• 75 ÷ 15 = 5
• 90 ÷ 15 = 6

Thus, the simplified form is 5/6.

Method 2: Prime Factorization Technique

Prime factorization is a more analytical method that involves breaking down both numbers into their most basic building blocks: prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11).

Factoring the Numerator (75)

• Is 75 divisible by 2? No (it is odd).
• Is 75 divisible by 3? Yes. 75 ÷ 3 = 25.
• Is 25 divisible by 3? No.
• Is 25 divisible by 5? Yes. 25 ÷ 5 = 5.
• Is 5 divisible by 5? Yes. 5 ÷ 5 = 1.

The prime factors of 75 are: 3 × 5 × 5.

Factoring the Denominator (90)

• Is 90 divisible by 2? Yes. 90 ÷ 2 = 45.
• Is 45 divisible by 2? No.
• Is 45 divisible by 3? Yes. 45 ÷ 3 = 15.
• Is 15 divisible by 3? Yes. 15 ÷ 3 = 5.
• Is 5 divisible by 5? Yes. 5 ÷ 5 = 1.

The prime factors of 90 are: 2 × 3 × 3 × 5.

Canceling Common Factors

Now, we write the fraction as the product of its prime factors: 75/90 = (3 × 5 × 5) / (2 × 3 × 3 × 5)

Identify the factors that appear in both the top and bottom:

• One "3" can be canceled from both.
• One "5" can be canceled from both.

What remains?

• Numerator: 5
• Denominator: 2 × 3 = 6

The result is 5/6.

Method 3: The Step-by-Step Reduction Method

For many, identifying the GCF of 15 immediately isn't intuitive. In such cases, the iterative or step-by-step reduction method is highly effective. This involves dividing the numerator and denominator by any common factor you can easily recognize until no more divisions are possible.

First Pass: Divide by 5

Since 75 ends in 5 and 90 ends in 0, both are clearly divisible by 5.

• 75 ÷ 5 = 15
• 90 ÷ 5 = 18
• New Fraction: 15/18

Second Pass: Divide by 3

Looking at 15 and 18, we can see that both are multiples of 3 (because 1+5=6 and 1+8=9, and both 6 and 9 are divisible by 3).

• 15 ÷ 3 = 5
• 18 ÷ 3 = 6
• New Fraction: 5/6

Since 5 is a prime number and not a factor of 6, no further simplification is possible. This method confirms the same result: 5/6.

Converting 75/90 into a Decimal

In many contexts, such as finance or scientific data, expressing a fraction as a decimal is more useful than its fraction form. To convert 75/90 (or its simplified version 5/6) into a decimal, we perform the division: 75 ÷ 90.

Using Long Division

1. Setup: Since 90 is larger than 75, the result will be less than 1. We place a 0 and a decimal point in our answer.
2. Add a Zero: Treat 75 as 75.0. How many times does 90 go into 750?
   • 90 × 8 = 720.
   • 750 - 720 = 30.
3. Next Step: Bring down another zero, making it 300. How many times does 90 go into 300?
   • 90 × 3 = 270.
   • 300 - 270 = 30.
4. Observe the Pattern: We again have a remainder of 30. Bringing down another zero makes it 300 again.
   • 90 × 3 = 270.
   • 300 - 270 = 30.

The Repeating Decimal

This process will repeat infinitely. Therefore, the decimal equivalent of 75/90 is 0.8333... (with the 3 repeating forever).

In mathematical notation, this is written as 0.83̅, where the bar over the 3 indicates it is a repeating digit.

Converting 75/90 into a Percentage

Percentages are fractions with a denominator of 100. To convert 75/90 into a percentage, you can use one of two main methods.

Method A: Multiplying the Decimal by 100

Take the decimal we just calculated and multiply by 100:

• 0.8333... × 100 = 83.333...%

Method B: The Proportion Method

Set up an equation where 75/90 is equal to X/100:

• (75 / 90) = (X / 100)
• Cross-multiply: 90X = 7500
• Divide by 90: X = 7500 ÷ 90
• X = 83.33...

Therefore, 75/90 is approximately 83.33%.

Understanding the Ratio 75:90

Fractions are closely related to ratios. The fraction 75/90 can be expressed as the ratio 75:90. Just like a fraction, a ratio can be simplified by dividing both sides by the same common factor.

• 75:90
• Divide both by 15: 5:6

This ratio tells us that for every 5 parts of the first quantity, there are 6 parts of the second. This is useful in chemistry (mixture proportions), cooking, or business (profit-sharing ratios).

Why Do We Simplify Fractions?

You might wonder why we bother simplifying 75/90 to 5/6 if the value remains exactly the same. There are several practical reasons:

1. Improved Clarity and Visualization

It is much easier for a human brain to visualize "5 out of 6" than "75 out of 90." When we simplify, we provide a clearer mental image of the proportion. If you are told you completed 5/6 of a project, you immediately understand you are nearly finished.

2. Standardized Communication

In the world of science, engineering, and education, using "lowest terms" is the standard. If every mathematician used different versions of the same fraction (e.g., 150/180, 750/900, 75/90), collaboration would become confusing. Simplifying to 5/6 ensures everyone is speaking the same numerical language.

3. Ease of Further Calculation

Imagine you need to multiply 75/90 by 12/25.

• Without simplification: (75 × 12) / (90 × 25) = 900 / 2250. Then you have to simplify 900/2250.
• With simplification: (5/6) × (12/25).
  • (5 ÷ 5) / (25 ÷ 5) = 1/5
  • (12 ÷ 6) / (6 ÷ 6) = 2/1
  • 1/1 × 2/5 = 2/5. Working with smaller numbers significantly reduces the risk of arithmetic errors.

Real-World Scenarios for 75/90

Academic Performance

The most common place a student encounters 75/90 is on a test or an assignment. If a student scores 75 out of 90 points:

• Their fractional score is 75/90.
• Their simplified proficiency is 5/6.
• Their percentage is 83.33% (usually a "B" or "A-" depending on the grading scale).

Time Management

If a task takes 75 minutes out of a 90-minute window, you have used 5/6 of the available time. This leaves 1/6 of the time (15 minutes) remaining.

Retail and Discounts

If a store offers an item originally priced at $90 for $75, the discount is $15.

• The sale price is 75/90 of the original price.
• The discount itself is 15/90, which simplifies to 1/6.
• Therefore, the store is offering roughly a 16.67% discount off the original price.

Common Mistakes to Avoid

When simplifying 75/90, students often make a few typical errors:

1. Dividing only the numerator: Some might divide 75 by 15 but forget to divide 90 by 15. This results in an incorrect value. Always perform the same operation on both the top and bottom.
2. Stopping too early: A student might divide by 5 and get 15/18, then assume it is finished. Always check if the new numerator and denominator share any more common factors.
3. Confusing factors with multiples: Factors are numbers that go into 75 and 90. Multiples are numbers that 75 and 90 go into. For simplification, we only care about factors.

Summary of 75/90 Conversions

The number 75/90 is a "proper fraction" because the numerator is smaller than the denominator. Its reduction to 5/6 reveals that it is a high-value proportion, representing over four-fifths of a whole.

Frequently Asked Questions

What is 75/90 simplified?

The fraction 75/90 simplified to its lowest terms is 5/6.

Is 75/90 an equivalent fraction to 150/180?

Yes. Equivalent fractions are different fractions that name the same amount. If you multiply the numerator and denominator of 75/90 by 2, you get 150/180. Therefore, they are equivalent.

How do you write 75/90 as a decimal?

To write 75/90 as a decimal, divide 75 by 90. The result is a repeating decimal: 0.8333...

What is the greatest common factor of 75 and 90?

The greatest common factor (GCF) of 75 and 90 is 15. This is the largest number that divides both 75 and 90 without a remainder.

Can 5/6 be simplified further?

No. Since 5 is a prime number and does not divide evenly into 6, the fraction 5/6 is already in its simplest form. The only common factor they share is 1.

Why is the decimal of 75/90 repeating?

A fraction results in a repeating decimal if the prime factors of its denominator (after the fraction is simplified) include numbers other than 2 or 5. Since the simplified denominator is 6, and its prime factors are 2 and 3, the presence of the 3 causes the decimal to repeat.

Final Conclusion

Simplifying 75/90 is a straightforward process once you identify the common factors shared by the numerator and denominator. By dividing both by 15, we arrive at the elegant and simple fraction of 5/6. Whether you are calculating grades, analyzing data, or simply solving a homework problem, understanding how to transition between 75/90, its simplified form, and its decimal or percentage counterparts is a vital mathematical competency. Through the methods of GCF, prime factorization, or iterative division, the result remains consistent, highlighting the beautiful logic and stability of mathematical principles.