Fractions form a fundamental part of mathematics and are crucial for various real-life applications. How Can I Learn Fractions effectively? This article, brought to you by LEARNS.EDU.VN, provides a step-by-step guide to mastering fractions, covering the basics, simplification, comparison, addition, subtraction, multiplication, division, and practical examples. Learning fractions involves understanding parts of a whole, equivalent forms, and performing arithmetic operations. Improve your mathematical proficiency and unlock a world of problem-solving skills. Dive into fraction concepts, fractional arithmetic, and essential math skills.
1. Understanding Fractions as Parts of a Whole
The fundamental concept of a fraction is that it represents a part of a whole. To truly understand this concept, we must delve into the components of a fraction, how they relate to each other, and how they visually represent portions of a whole.
1.1. Defining Numerator and Denominator
A fraction consists of two primary parts: the numerator and the denominator. The denominator is the bottom number, indicating the total number of equal parts into which the whole is divided. The numerator is the top number, specifying how many of these parts are being considered.
The fraction 3/4 (three-fourths) means that a whole is divided into four equal parts, and we are referring to three of those parts. Understanding this distinction is key to grasping fractions.
1.2. Visual Representation of Fractions
Visual aids are powerful tools for understanding fractions. Imagine a pie divided into equal slices. If the pie is cut into 6 equal slices, each slice represents 1/6 (one-sixth) of the pie. If you take 2 slices, you have 2/6 (two-sixths) of the pie.
Using diagrams such as pie charts, bar models, or even real-world objects can make the concept of fractions more tangible. By seeing how fractions break down a whole, learners can better comprehend their values and relationships.
1.3. Fractions in Everyday Life
Fractions are not just abstract mathematical concepts; they are prevalent in daily life. Consider these scenarios:
- Cooking: Recipes often use fractions. For example, a recipe might call for 1/2 cup of flour or 1/4 teaspoon of salt.
- Time: An hour is divided into 60 minutes. Therefore, 30 minutes is 1/2 hour.
- Shopping: Sales are often expressed as fractions or percentages, such as 1/3 off the original price.
- Sports: In basketball, a player might make 2/5 of their free throws.
- Measurements: Measuring lengths, weights, or volumes often involves fractions, such as 1/8 inch or 3/4 pound.
Recognizing these examples helps learners see the practical relevance of fractions, making the topic more engaging and less intimidating.
2. Simplifying Fractions: Reducing to Lowest Terms
Simplifying fractions means reducing them to their simplest form. This makes the fraction easier to understand and work with. The process involves finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.
2.1. Identifying the Greatest Common Factor (GCF)
The greatest common factor (GCF) is the largest number that divides both the numerator and the denominator evenly. For example, consider the fraction 6/12. The factors of 6 are 1, 2, 3, and 6. The factors of 12 are 1, 2, 3, 4, 6, and 12. The largest number that appears in both lists is 6, so the GCF of 6 and 12 is 6.
2.2. Dividing by the GCF
Once you’ve found the GCF, divide both the numerator and the denominator by that number. In our example, 6/12, the GCF is 6. So, we divide both the numerator and the denominator by 6:
- 6 ÷ 6 = 1
- 12 ÷ 6 = 2
Thus, the simplified form of 6/12 is 1/2.
2.3. Step-by-Step Simplification
Let’s break down the process into clear steps:
- List the factors of both the numerator and the denominator.
- Identify the GCF.
- Divide both the numerator and the denominator by the GCF.
- Write the new fraction in its simplest form.
For instance, simplify 12/18:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- GCF: 6
- 12 ÷ 6 = 2
- 18 ÷ 6 = 3
- Simplified fraction: 2/3
2.4. Why Simplifying Matters
Simplifying fractions is crucial for several reasons:
- Easier to understand: Simplified fractions are easier to visualize and compare.
- Reduced calculations: Working with smaller numbers simplifies arithmetic operations.
- Standard form: Simplified fractions are often required in standardized tests and academic settings.
By consistently simplifying fractions, you build a stronger foundation in mathematics.
3. Comparing Fractions: Determining Relative Values
Comparing fractions involves determining which fraction is greater, lesser, or equal to another. This skill is essential for making informed decisions in various contexts, from cooking to finance.
3.1. Common Denominators
The easiest way to compare fractions is to ensure they have a common denominator. This means both fractions have the same number as the denominator. To achieve this, find the least common multiple (LCM) of the denominators.
For example, to compare 1/3 and 2/5, first, find the LCM of 3 and 5, which is 15. Then, convert both fractions to have a denominator of 15:
- 1/3 = (1 x 5) / (3 x 5) = 5/15
- 2/5 = (2 x 3) / (5 x 3) = 6/15
Now, it’s easy to see that 6/15 is greater than 5/15, so 2/5 is greater than 1/3.
3.2. Cross-Multiplication
An alternative method is cross-multiplication. To compare two fractions, a/b and c/d, multiply the numerator of the first fraction by the denominator of the second (a x d) and the numerator of the second fraction by the denominator of the first (c x b).
Compare the results:
- If a x d > c x b, then a/b > c/d
- If a x d < c x b, then a/b < c/d
- If a x d = c x b, then a/b = c/d
Using this method for 1/3 and 2/5:
- 1 x 5 = 5
- 2 x 3 = 6
- Since 5 < 6, 1/3 < 2/5
3.3. Benchmarking
Benchmarking involves comparing fractions to a common reference point, often 1/2. This is particularly useful for quick, approximate comparisons.
For example:
- 3/7: Is 3/7 more or less than 1/2? Since 3 is less than half of 7, 3/7 < 1/2.
- 5/8: Is 5/8 more or less than 1/2? Since 5 is more than half of 8, 5/8 > 1/2.
Using benchmarks can simplify comparisons, especially when exact values are not necessary.
3.4. Real-World Comparisons
Consider scenarios where comparing fractions is essential:
- Shopping: Comparing discounts, such as 1/3 off versus 1/4 off.
- Cooking: Adjusting recipe quantities, such as doubling a recipe that calls for 2/3 cup of sugar.
- Investing: Evaluating investment returns, such as comparing a 1/10 return to a 1/8 return.
4. Adding Fractions: Combining Portions
Adding fractions involves combining two or more fractional quantities into a single fraction. This skill is crucial in many real-life situations, such as calculating total measurements or combining ingredients in a recipe.
4.1. Fractions with Common Denominators
The simplest case of adding fractions is when they have the same denominator. To add fractions with common denominators, add the numerators and keep the denominator the same.
For example, to add 2/7 and 3/7:
- 2/7 + 3/7 = (2 + 3) / 7 = 5/7
This means you’re combining two parts and three parts of a whole that is divided into seven parts.
4.2. Fractions with Different Denominators
Adding fractions with different denominators requires finding a common denominator first. The least common multiple (LCM) is the most efficient choice. Once you have the common denominator, convert each fraction to an equivalent fraction with that denominator, and then add the numerators.
For example, to add 1/3 and 2/5:
- Find the LCM of 3 and 5, which is 15.
- Convert each fraction:
- 1/3 = (1 x 5) / (3 x 5) = 5/15
- 2/5 = (2 x 3) / (5 x 3) = 6/15
- Add the fractions:
- 5/15 + 6/15 = (5 + 6) / 15 = 11/15
4.3. Adding Mixed Numbers
A mixed number is a whole number combined with a fraction, such as 1 1/2. To add mixed numbers, you can either convert them to improper fractions or add the whole numbers and fractions separately.
Method 1: Convert to Improper Fractions
- Convert each mixed number to an improper fraction.
- Find a common denominator if necessary.
- Add the fractions.
- Simplify or convert back to a mixed number if needed.
For example, add 1 1/2 and 2 1/4:
- Convert to improper fractions:
- 1 1/2 = (1 x 2 + 1) / 2 = 3/2
- 2 1/4 = (2 x 4 + 1) / 4 = 9/4
- Find a common denominator: The LCM of 2 and 4 is 4.
- 3/2 = (3 x 2) / (2 x 2) = 6/4
- Add the fractions:
- 6/4 + 9/4 = 15/4
- Convert back to a mixed number:
- 15/4 = 3 3/4
Method 2: Add Whole Numbers and Fractions Separately
- Add the whole numbers.
- Add the fractions (finding a common denominator if necessary).
- Combine the results.
Using the same example, 1 1/2 and 2 1/4:
- Add whole numbers:
- 1 + 2 = 3
- Add fractions:
- 1/2 + 1/4 = 2/4 + 1/4 = 3/4
- Combine the results:
- 3 + 3/4 = 3 3/4
4.4. Practical Applications
Adding fractions is useful in many real-world scenarios:
- Cooking: Combining ingredient amounts, such as adding 1/2 cup of flour and 1/4 cup of sugar.
- Construction: Calculating lengths, such as adding pieces of wood that are 2 1/2 feet and 3 1/4 feet long.
- Time management: Adding time intervals, such as working for 1/3 of an hour and then taking a 1/6 of an hour break.
5. Subtracting Fractions: Finding the Difference
Subtracting fractions involves finding the difference between two fractional quantities. This skill is essential in various situations, such as determining remaining amounts or calculating changes.
5.1. Fractions with Common Denominators
The simplest case of subtracting fractions is when they have the same denominator. To subtract fractions with common denominators, subtract the numerators and keep the denominator the same.
For example, to subtract 2/5 from 4/5:
- 4/5 – 2/5 = (4 – 2) / 5 = 2/5
5.2. Fractions with Different Denominators
Subtracting fractions with different denominators requires finding a common denominator first. The least common multiple (LCM) is the most efficient choice. Once you have the common denominator, convert each fraction to an equivalent fraction with that denominator, and then subtract the numerators.
For example, to subtract 1/4 from 2/3:
- Find the LCM of 3 and 4, which is 12.
- Convert each fraction:
- 2/3 = (2 x 4) / (3 x 4) = 8/12
- 1/4 = (1 x 3) / (4 x 3) = 3/12
- Subtract the fractions:
- 8/12 – 3/12 = (8 – 3) / 12 = 5/12
5.3. Subtracting Mixed Numbers
To subtract mixed numbers, you can either convert them to improper fractions or subtract the whole numbers and fractions separately.
Method 1: Convert to Improper Fractions
- Convert each mixed number to an improper fraction.
- Find a common denominator if necessary.
- Subtract the fractions.
- Simplify or convert back to a mixed number if needed.
For example, subtract 1 1/4 from 3 1/2:
- Convert to improper fractions:
- 3 1/2 = (3 x 2 + 1) / 2 = 7/2
- 1 1/4 = (1 x 4 + 1) / 4 = 5/4
- Find a common denominator: The LCM of 2 and 4 is 4.
- 7/2 = (7 x 2) / (2 x 2) = 14/4
- Subtract the fractions:
- 14/4 – 5/4 = 9/4
- Convert back to a mixed number:
- 9/4 = 2 1/4
Method 2: Subtract Whole Numbers and Fractions Separately
- Subtract the whole numbers.
- Subtract the fractions (finding a common denominator if necessary).
- Combine the results. If the fraction in the second mixed number is larger than the fraction in the first, you may need to borrow from the whole number.
Using the same example, 3 1/2 – 1 1/4:
- Subtract whole numbers:
- 3 – 1 = 2
- Subtract fractions:
- 1/2 – 1/4 = 2/4 – 1/4 = 1/4
- Combine the results:
- 2 + 1/4 = 2 1/4
5.4. Subtracting from a Whole Number
To subtract a fraction from a whole number, rewrite the whole number as a fraction with the same denominator as the fraction you are subtracting.
For example, subtract 2/7 from 3:
- Rewrite 3 as a fraction with a denominator of 7:
- 3 = 3/1 = (3 x 7) / (1 x 7) = 21/7
- Subtract the fractions:
- 21/7 – 2/7 = (21 – 2) / 7 = 19/7
- Convert back to a mixed number if needed:
- 19/7 = 2 5/7
5.5. Practical Applications
Subtracting fractions is useful in many real-world scenarios:
- Cooking: Determining the remaining amount of an ingredient after using a portion, such as starting with 3/4 cup of sugar and using 1/3 cup.
- Construction: Calculating the length of material left after cutting a piece, such as starting with a 5 1/2 feet long board and cutting off 2 1/4 feet.
- Finance: Calculating remaining balances, such as starting with $100 and spending 1/4 of it.
6. Multiplying Fractions: Scaling Portions
Multiplying fractions involves finding a fraction of another fraction or a fraction of a whole number. This skill is vital for scaling recipes, calculating areas, and understanding proportions.
6.1. Multiplying Two Fractions
To multiply two fractions, multiply the numerators together and multiply the denominators together.
For example, to multiply 2/5 by 3/4:
- (2/5) x (3/4) = (2 x 3) / (5 x 4) = 6/20
Then, simplify the resulting fraction if possible:
- 6/20 = 3/10
6.2. Multiplying a Fraction by a Whole Number
To multiply a fraction by a whole number, treat the whole number as a fraction with a denominator of 1. Then, multiply the numerators and the denominators.
For example, to multiply 3/8 by 5:
- (3/8) x (5/1) = (3 x 5) / (8 x 1) = 15/8
Convert the improper fraction to a mixed number if needed:
- 15/8 = 1 7/8
6.3. Multiplying Mixed Numbers
To multiply mixed numbers, convert them to improper fractions first, and then multiply the fractions as described above.
For example, to multiply 1 1/2 by 2 2/3:
- Convert to improper fractions:
- 1 1/2 = (1 x 2 + 1) / 2 = 3/2
- 2 2/3 = (2 x 3 + 2) / 3 = 8/3
- Multiply the fractions:
- (3/2) x (8/3) = (3 x 8) / (2 x 3) = 24/6
- Simplify the result:
- 24/6 = 4
6.4. Practical Applications
Multiplying fractions is useful in various real-world scenarios:
- Cooking: Scaling recipes, such as halving a recipe that calls for 2/3 cup of flour.
- Construction: Calculating areas, such as finding the area of a rectangular garden that is 3/4 meter wide and 2/5 meter long.
- Finance: Calculating discounts, such as finding the sale price of an item that is 1/3 off the original price of $60.
7. Dividing Fractions: Sharing Portions Equally
Dividing fractions involves determining how many times one fraction fits into another. This skill is crucial for solving problems related to sharing, measuring, and scaling.
7.1. Dividing Two Fractions
To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
For example, to divide 2/3 by 3/4:
- Find the reciprocal of 3/4:
- Reciprocal of 3/4 = 4/3
- Multiply 2/3 by 4/3:
- (2/3) ÷ (3/4) = (2/3) x (4/3) = (2 x 4) / (3 x 3) = 8/9
7.2. Dividing a Fraction by a Whole Number
To divide a fraction by a whole number, treat the whole number as a fraction with a denominator of 1, then multiply the fraction by the reciprocal of the whole number.
For example, to divide 3/5 by 4:
- Rewrite 4 as 4/1.
- Find the reciprocal of 4/1:
- Reciprocal of 4/1 = 1/4
- Multiply 3/5 by 1/4:
- (3/5) ÷ (4/1) = (3/5) x (1/4) = (3 x 1) / (5 x 4) = 3/20
7.3. Dividing Mixed Numbers
To divide mixed numbers, convert them to improper fractions first, and then divide the fractions as described above.
For example, to divide 1 1/3 by 2 1/2:
- Convert to improper fractions:
- 1 1/3 = (1 x 3 + 1) / 3 = 4/3
- 2 1/2 = (2 x 2 + 1) / 2 = 5/2
- Find the reciprocal of 5/2:
- Reciprocal of 5/2 = 2/5
- Multiply 4/3 by 2/5:
- (4/3) ÷ (5/2) = (4/3) x (2/5) = (4 x 2) / (3 x 5) = 8/15
7.4. Practical Applications
Dividing fractions is useful in various real-world scenarios:
- Cooking: Dividing a quantity into equal portions, such as dividing 2/3 cup of sugar into 4 equal portions.
- Construction: Determining how many pieces of a certain length can be cut from a longer piece, such as cutting 1/4 meter pieces from a 3/4 meter long board.
- Finance: Dividing profits or losses, such as splitting 1/2 of a business’s profits equally among 3 partners.
8. Practical Examples of Using Fractions
Fractions are not abstract concepts; they are practical tools that help us solve real-world problems every day. Understanding how to apply fractions in different scenarios can make them more relatable and easier to master.
8.1. Sharing a Pizza
Imagine you and three friends are sharing a pizza. The pizza is cut into 8 slices. You want to know what fraction of the pizza each person gets.
- There are 4 people sharing the pizza.
- The pizza has 8 slices.
- Each person gets 8/4 = 2 slices.
- The fraction of the pizza each person gets is 2/8, which simplifies to 1/4.
So, each person gets 1/4 of the pizza.
8.2. Measuring Ingredients
You are baking a cake, and the recipe calls for 2/3 cup of flour. You only want to make half of the recipe. How much flour do you need?
- The original recipe calls for 2/3 cup of flour.
- You want to make 1/2 of the recipe.
- Multiply 2/3 by 1/2: (2/3) x (1/2) = 2/6
- Simplify the fraction: 2/6 = 1/3
You need 1/3 cup of flour.
8.3. Calculating Time
You spend 1/4 of your day at school, 1/8 of your day doing homework, and the rest of your day doing other activities. How much of your day is spent on other activities?
- A full day is represented by 1.
- Time at school: 1/4
- Time doing homework: 1/8
- Add the fractions: 1/4 + 1/8 = 2/8 + 1/8 = 3/8
- Subtract this from the whole day: 1 – 3/8 = 8/8 – 3/8 = 5/8
You spend 5/8 of your day on other activities.
8.4. Determining Distance
You have to travel 2/5 of a journey. You have already completed 1/3 of the total distance. How much further do you have to travel?
- Total distance to travel: 2/5
- Distance already traveled: 1/3
- Subtract the distances: 2/5 – 1/3
- Find a common denominator: 2/5 = 6/15, 1/3 = 5/15
- Subtract: 6/15 – 5/15 = 1/15
You have 1/15 of the journey left to travel.
8.5. Managing Finances
You have $60 and spend 2/3 of it on groceries. How much money did you spend on groceries?
- Total amount of money: $60
- Fraction spent on groceries: 2/3
- Multiply $60 by 2/3: (2/3) x $60 = $120/3 = $40
You spent $40 on groceries.
8.6. Construction Projects
You need to cut a wooden plank into pieces that are each 1/4 meter long. If the plank is 3/4 meter long, how many pieces can you cut?
- Total length of the plank: 3/4 meter
- Length of each piece: 1/4 meter
- Divide the total length by the length of each piece: (3/4) ÷ (1/4) = (3/4) x (4/1) = 12/4
- Simplify: 12/4 = 3
You can cut 3 pieces from the plank.
8.7. Scaling Recipes
A recipe for cookies calls for 1 1/2 cups of sugar. You want to make double the recipe. How much sugar do you need?
- Original amount of sugar: 1 1/2 cups
- Convert to an improper fraction: 1 1/2 = 3/2
- Multiply by 2 to double the recipe: (3/2) x 2 = 6/2
- Simplify: 6/2 = 3
You need 3 cups of sugar.
9. Tips and Tricks for Mastering Fractions
Mastering fractions can be challenging, but with the right approach and strategies, anyone can become proficient. Here are some essential tips and tricks to help you succeed.
9.1. Start with the Basics
Before tackling complex operations, ensure you have a solid understanding of the basic concepts. Know what numerators and denominators represent, and understand how fractions relate to a whole.
9.2. Use Visual Aids
Visual aids can make abstract concepts more concrete. Use diagrams, such as pie charts, bar models, and number lines, to visualize fractions and their relationships.
9.3. Practice Regularly
Consistent practice is key to mastering any mathematical skill. Set aside time each day to work on fraction problems. Start with simple exercises and gradually increase the difficulty as you improve.
9.4. Understand Equivalent Fractions
Recognize that different fractions can represent the same value. For example, 1/2, 2/4, and 3/6 are all equivalent fractions. Understanding this concept can simplify many operations.
9.5. Master Simplification
Simplifying fractions makes them easier to work with. Practice finding the greatest common factor (GCF) and dividing both the numerator and denominator by it.
9.6. Learn Common Denominators
When adding or subtracting fractions, knowing how to find a common denominator quickly is essential. Practice finding the least common multiple (LCM) of different numbers.
9.7. Break Down Complex Problems
Complex fraction problems can be overwhelming. Break them down into smaller, more manageable steps. This makes the problem less intimidating and easier to solve.
9.8. Use Real-World Examples
Relate fractions to real-world situations to make them more meaningful. Cooking, measuring, and financial calculations all involve fractions.
9.9. Utilize Online Resources
There are numerous online resources available to help you learn and practice fractions. Websites like learns.edu.vn offer tutorials, exercises, and interactive tools.
9.10. Seek Help When Needed
Don’t hesitate to ask for help if you’re struggling with fractions. Teachers, tutors, and online forums can provide valuable assistance.
9.11. Stay Positive
Learning fractions takes time and effort. Stay positive and persistent, and celebrate your progress along the way.
9.12. Make it Fun
Turn learning fractions into a game. Use flashcards, puzzles, and online games to make the process more enjoyable.
10. Advanced Fraction Concepts
Once you’ve mastered the basic operations with fractions, you can explore more advanced concepts that build upon this foundation. These concepts are essential for higher-level mathematics and various real-world applications.
10.1. Complex Fractions
A complex fraction is a fraction where the numerator, the denominator, or both contain fractions. To simplify a complex fraction, multiply the numerator and denominator by the least common denominator (LCD) of all fractions within the complex fraction.
For example, consider the complex fraction (1/2) / (2/3):
- Identify the fractions within the complex fraction: 1/2 and 2/3.
- Find the LCD of the denominators 2 and 3, which is 6.
- Multiply both the numerator and the denominator by 6:
- [(1/2) x 6] / [(2/3) x 6] = 3 / 4
10.2. Ratios and Proportions
Fractions are closely related to ratios and proportions. A ratio compares two quantities, while a proportion states that two ratios are equal.
For example, if the ratio of apples to oranges is 3:4, it means that for every 3 apples, there are 4 oranges. This can be expressed as the fraction 3/4.
Proportions can be used to solve problems involving scaling and comparison. For example, if 2 apples cost $1, how much will 6 apples cost?
- Set up a proportion: 2/1 = 6/x
- Cross-multiply: 2x = 6
- Solve for x: x = 3
Therefore, 6 apples will cost $3.
10.3. Fractions and Percentages
Fractions and percentages are different ways of representing the same values. To convert a fraction to a percentage, multiply the fraction by 100.
For example, to convert 3/4 to a percentage:
- (3/4) x 100 = 75%
Conversely, to convert a percentage to a fraction, divide the percentage by 100 and simplify the fraction.
For example, to convert 25% to a fraction:
- 25/100 = 1/4
10.4. Scientific Notation
Fractions are used in scientific notation to express very large or very small numbers. Scientific notation represents a number as a product of a number between 1 and 10 and a power of 10.
For example, the number 0.0005 can be written as 5 x 10^-4. The exponent -4 indicates that the decimal point should be moved four places to the left.
10.5. Fractions in Algebra
Fractions are essential in algebra for solving equations and simplifying expressions. Algebraic fractions involve variables in the numerator, denominator, or both.
For example, consider the equation:
- (x/2) + (x/3) = 5
To solve for x, find a common denominator and combine the fractions:
- Common denominator: 6
- (3x/6) + (2x/6) = 5
- (5x/6) = 5
- Multiply both sides by 6: 5x = 30
- Divide by 5: x = 6
Fractions are also used in algebraic expressions, such as simplifying rational expressions.
10.6. Calculus and Fractions
In calculus, fractions are used extensively in the study of limits, derivatives, and integrals. Understanding how to manipulate fractions is crucial for solving calculus problems.
For example, consider the limit:
- lim (x→2) (x^2 – 4) / (x – 2)
To evaluate this limit, factor the numerator and simplify the fraction:
- (x^2 – 4) = (x – 2)(x + 2)
- (x – 2)(x + 2) / (x – 2) = x + 2
- lim (x→2) (x + 2) = 2 + 2 = 4
10.7. Practical Engineering
Fractions are important for practical engineering applications such as calculating measurements, tolerances, and material properties.
| Application | Description | Example |
|---|---|---|
| Measurements | Fractions are used for precise measurements in construction, manufacturing, and other engineering fields. | A blueprint specifies a bolt length of 3 1/4 inches. |
| Tolerances | Tolerances define the acceptable range of variation in measurements. They are often expressed as fractions or percentages. | A mechanical part has a tolerance of ± 1/16 inch. |
| Material Properties | Material properties, such as density and strength, are often expressed as fractions or ratios. These properties are used to calculate the performance of engineering structures. | The density of steel is approximately 7 4/5 grams per cubic centimeter. |
