Understanding decimals and fractions is an essential skill in mathematics. In Lesson 1, we explored the relationship between decimals and fractions and practiced converting between the two. This homework assignment provided students with the opportunity to reinforce their understanding and apply the concepts learned in class.
The answer key for Lesson 1 homework practice on decimals and fractions is a valuable resource for students and teachers alike. It allows students to check their work and identify any errors they may have made. It also serves as a guide for teachers in assessing student comprehension and providing feedback on their performance.
By reviewing the answer key, students can gain a deeper understanding of the concepts covered in the lesson. They can see how decimals and fractions are related and how to convert between them accurately. This knowledge is crucial for future math assignments and will help students build a strong foundation for more advanced topics.
Lesson 1 Homework Practice: Decimals and Fractions Answer Key
Below is the answer key for the homework practice on decimals and fractions. Use this as a guide to check your work and understand any mistakes you may have made.
1. Convert the following fractions to decimals:
- 1/4 = 0.25
- 3/5 = 0.6
- 2/3 = 0.67
- 5/8 = 0.625
- 7/10 = 0.7
2. Write the following decimals as fractions in simplest form:
- 0.6 = 3/5
- 0.75 = 3/4
- 0.4 = 2/5
- 0.125 = 1/8
- 0.85 = 17/20
Remember to always check your answers by converting between decimals and fractions using the appropriate method. Practice these conversions regularly to improve your understanding and speed.
Reviewing Decimals and Fractions
Decimals and fractions are mathematical concepts that represent numbers which fall between whole numbers. Understanding how to convert between decimals and fractions is important in various areas of life, including finance, measurement, and problem-solving.
Decimals: A decimal is a number that includes a decimal point, which separates the whole number from the decimal portion. The decimal portion represents a fraction of a whole number, with each digit after the decimal point representing a specific value. For example, in the decimal 0.25, the 2 represents two tenths (2/10) and the 5 represents five hundredths (5/100).
Fractions: A fraction is a way to express a part of a whole number or a ratio between two numbers. It consists of two numbers separated by a slash (“/”). The number above the slash is called the numerator, and the number below the slash is called the denominator. For example, in the fraction 3/4, the 3 represents three parts out of four equal parts.
Converting between decimals and fractions can be done by understanding the relationship between the two. To convert a decimal to a fraction, we can use the place value of the decimal digits. For example, the decimal 0.75 can be converted to the fraction 75/100, which can be simplified to 3/4. To convert a fraction to a decimal, we can divide the numerator by the denominator. For example, the fraction 1/2 can be converted to the decimal 0.5.
Understanding and being able to work with decimals and fractions is an important skill in mathematics. It allows us to easily compare and manipulate numbers, and provides a foundation for more advanced mathematical concepts.
Understanding Place Value of Decimals
When working with decimals, it is crucial to understand the concept of place value. Place value refers to the value of a digit based on its position in a number. In decimal numbers, the position of a digit to the right of the decimal point determines its place value.
Decimal Places: The number of digits to the right of the decimal point determines the decimal places. Each digit represents a specific place value, with the first digit to the right of the decimal point being in the tenths place, the second digit in the hundredths place, the third digit in the thousandths place, and so on.
For example, in the decimal number 0.123, the digit 2 is in the hundredths place because it is the second digit to the right of the decimal point. Similarly, in the decimal number 15.6789, the digit 7 is in the thousandths place because it is the fourth digit to the right of the decimal point.
Comparing Decimal Numbers: When comparing decimal numbers, it’s important to start from the left and compare the digits in each place value. Begin by comparing the digits to the left of the decimal point, and if they are equal, move to the next place value to the right. Continue this process until a difference is found.
Rounding Decimals: Rounding decimals involves changing a decimal to a number that has fewer decimal places. To round a decimal, identify the place value you want to round to. Look at the digit to the right of the desired place value. If the digit is 5 or greater, round up by increasing the digit in the desired place value by 1. If the digit is less than 5, round down by leaving the digit in the desired place value as is.
Understanding place value of decimals is essential for performing operations such as addition, subtraction, multiplication, and division accurately. It helps in correctly placing the digits and maintaining the correct precision in calculations.
Converting Fractions to Decimals
When working with fractions, it is often useful to convert them into decimal form. This allows us to compare and perform calculations more easily. To convert a fraction to a decimal, we divide the numerator (the top number) by the denominator (the bottom number).
Let’s take an example. Say we have the fraction 3/4. To convert this to a decimal, we divide 3 by 4. The quotient is 0.75. Therefore, 3/4 is equal to 0.75 as a decimal.
It is important to note that not all fractions have exact decimal equivalents. Some fractions, when converted to decimals, result in repeating or terminating decimals. For example, the fraction 1/3 cannot be expressed as an exact decimal and instead, it becomes 0.3333… where the 3s repeat infinitely.
To convert a fraction to a decimal, we can use long division, a calculator, or a conversion chart. Long division involves dividing the numerator by the denominator and continuing the division until we either reach a terminating decimal or notice a repeating pattern in the decimal digits.
When working with decimals, it’s important to understand place value and rounding. The decimal point separates the whole number part from the fractional part. The digits to the right of the decimal point represent parts of a whole, such as tenths, hundredths, and so on. When rounding decimals, we look at the digit to the right of the desired decimal place and determine if it should be rounded up or down based on the digit following it.
In summary, converting fractions to decimals is an essential skill in mathematics. It allows for easier comparison and calculation. Understanding the concepts of place value and rounding will help in accurately representing fractions as decimals.
Converting Decimals to Fractions
Converting decimals to fractions is an important skill in mathematics. It allows us to represent decimal numbers as fractions, which can sometimes be easier to work with. To convert a decimal to a fraction, we need to understand the place value of the decimal and its relationship to the fraction.
First, let’s review the place value of decimals. A decimal consists of two parts: the whole number part and the decimal part. The decimal part is divided into tenths, hundredths, thousandths, and so on, depending on the number of decimal places. For example, in the decimal 0.75, the 7 is in the tenths place and the 5 is in the hundredths place.
To convert a decimal to a fraction, we can use the place value of the decimal. We write the decimal as the numerator of the fraction and the place value as the denominator. For example, to convert 0.75 to a fraction, we write 75 as the numerator and 100 as the denominator, because the 7 is in the tenths place and the 5 is in the hundredths place. Therefore, the fraction equivalent of 0.75 is 75/100.
In some cases, we may need to simplify the fraction by finding the greatest common factor of the numerator and denominator and dividing both by it. For example, if we simplify 75/100, we can divide both the numerator and denominator by 25 to get 3/4, which is the simplest form of the fraction.
It is important to practice converting decimals to fractions to develop fluency in this skill. With regular practice, we can quickly and accurately convert decimals to fractions, allowing us to solve mathematical problems more efficiently.
Adding and Subtracting Decimals
Adding and subtracting decimals is an important skill in mathematics. Decimals are used to represent numbers between whole numbers, and they allow for more precise calculations. When adding or subtracting decimals, it is crucial to align the decimal points before performing the operation.
To add decimals, start by aligning the decimal points of the numbers being added. Then, add the numbers as if they were whole numbers, placing the decimal point in the sum directly below the aligned decimal points. It is important to remember to carry any values that exceed 9 when adding the decimal places.
For example, let’s add 2.7 and 3.85. By aligning the decimal points, we get:
- 2.70
- + 3.85
- __
- 6.55
Subtracting decimals follows a similar process. Again, align the decimal points and subtract the numbers as if they were whole numbers. Place the decimal point in the difference directly below the aligned decimal points. If necessary, borrow from the whole number places when subtracting the decimal places.
For example, let’s subtract 4.91 from 7.35. By aligning the decimal points, we get:
- 7.35
- – 4.91
- __
- 2.44
Remember to practice adding and subtracting decimals to build your skills and improve accuracy in mathematical operations involving decimals. With practice, you will become more comfortable and proficient in working with decimals.
Multiplying Decimals
When multiplying decimals, it is important to remember a few key steps. First, take the two decimal numbers and ignore the decimal point for now. Treat them as whole numbers and multiply them as you normally would. After getting the product, count the total number of decimal places in both numbers. This will determine the placement of the decimal point in the final answer.
To illustrate this process, let’s look at an example. Suppose we want to multiply 0.25 by 0.5. Ignoring the decimal point, we have 25 multiplied by 5, which equals 125. Now, both numbers have a total of two decimal places. Since the product has a total of two decimal places as well, we can place the decimal point in the answer: 0.125.
It is important to pay attention to the number of decimal places when multiplying decimals. If one number has two decimal places and the other has three, the final product will have a total of five decimal places. Similarly, if one number has no decimal places and the other has four, the final product will have a total of four decimal places. Remembering to count the decimal places will help ensure the accuracy of your calculations.
When multiplying decimals, it is a good practice to double-check your answer and compare it to an estimation. If your estimated answer is significantly different from the actual product, it is a sign that you may have made a mistake in your calculation. Revisit the steps and identify any errors to correct them.
In conclusion, multiplying decimals requires following a specific process. By treating the decimals as whole numbers, multiplying them, and correctly placing the decimal point in the final answer based on the number of decimal places, you can perform accurate calculations. Remember to double-check your work and compare it to an estimation to ensure accuracy.
Dividing Decimals
Dividing decimals involves dividing one decimal number by another. It is similar to dividing whole numbers, but with some additional steps to account for the decimal places.
To divide decimals, follow these steps:
- Write the divisor and dividend in standard division format, with the decimal point lined up.
- If necessary, add zeros to the right of the dividend to make it a whole number.
- Divide as you would with whole numbers, ignoring the decimal point.
- Count the number of decimal places in the dividend.
- Place the decimal point directly above the decimal point in the divisor.
- Add zeros to the right of the dividend if needed, and continue dividing until you have the desired number of decimal places in the quotient.
For example, let’s divide 4.5 by 0.5:
| 9 | | | 4.5 | |
| . | 0 | 5 |
In this example, we write 4.5 as the dividend and 0.5 as the divisor. We add zeros to the right of the dividend, making it 45. Then we divide 45 by 5, which equals 9. Finally, we place the decimal point directly above the decimal point in the divisor and add zeros to the right of the dividend, resulting in a quotient of 9.0.
Remember to always check your final answer by multiplying the quotient by the divisor to see if the result equals the dividend.
Applying Decimals and Fractions in Real-Life Situations
Decimals and fractions are used in many real-life situations, from calculating the cost of groceries to determining the length of time for a recipe to cook. These mathematical concepts are essential for everyday tasks and can help individuals make informed decisions based on accurate calculations.
One common application of decimals and fractions is in financial transactions. When making a purchase, individuals often encounter prices with decimal values, such as $4.99 or $3.75. Understanding decimals allows people to calculate the total cost of their purchase, including taxes, discounts, and any additional fees or charges.
For example:
- If a person wants to buy two items that cost $4.99 each, they must calculate the total cost by multiplying the price by the quantity: $4.99 x 2 = $9.98.
- Alternatively, if there is a 10% discount on each item, the individual can calculate the discounted price by multiplying the original price by the discount rate: $4.99 x 0.10 = $0.499, then subtract that amount from the original price: $4.99 – $0.499 = $4.491.
Another practical use of decimals and fractions is in cooking and baking. Many recipes require precise measurements, such as 1/2 cup of flour or 1.5 teaspoons of baking powder. Understanding fractions and decimals allows individuals to accurately measure and combine ingredients, ensuring the desired outcome of the recipe.
For example:
- If a recipe calls for 1/2 cup of flour and an individual wants to make a double batch, they need to calculate the new measurement by multiplying the original measurement by the desired quantity: 1/2 x 2 = 1 cup of flour.
- Similarly, if a recipe calls for 2.5 teaspoons of baking powder and an individual only has a 1/2 teaspoon measure, they can calculate the number of measurements needed by dividing the required amount by the available measure: 2.5 / 0.5 = 5. They will need to use the 1/2 teaspoon measure five times to achieve the correct quantity.
By understanding and applying decimals and fractions in real-life situations, individuals can become more adept at making accurate calculations and informed decisions. These mathematical concepts are invaluable tools that can be used in various aspects of everyday life, from personal finance to culinary endeavors.