DFP Conversions: Decimals, Fractions, and Percents

Have you ever seen a sale advertised as "25% off"? Or perhaps you've heard someone talk about "half a pizza"? And what about batting averages like ".300" in baseball? All of these are different ways to talk about parts of a whole, and they use fractions, decimals, and percentages!

These three forms are like different languages that all say the same thing. Learning to translate between them is a super important math skill that you'll use all the time, both in school and in real life.

What Are They?

• Fraction: A fraction represents a part of a whole, like a slice of pizza. It's written as one number over another (e.g., 1/2, 3/4, 5/8).
• Decimal: A decimal also represents a part of a whole, but it uses place value with a decimal point (e.g., 0.5, 0.75, 0.625).
• Percent: A percent means "out of one hundred." It's a special kind of ratio that compares a number to 100, and it uses the percent symbol (%) (e.g., 50%, 75%, 62.5%).

How to Convert Between Them: Your Translation Guide!

1. Fraction to Decimal

To change a fraction to a decimal, you simply divide the top number (numerator) by the bottom number (denominator).

Example: Convert 3/4 to a decimal.

• 3 ÷ 4 = 0.75

2. Decimal to Percent

To change a decimal to a percent, multiply the decimal by 100 and add a percent sign (%). You can also think of this as moving the decimal point two places to the right.

Example: Convert 0.75 to a percent.

• 0.75 × 100 = 75
• Add the % sign: 75%

3. Percent to Decimal

To change a percent to a decimal, remove the percent sign (%) and divide the number by 100. You can also think of this as moving the decimal point two places to the left.

Example: Convert 75% to a decimal.

• Remove %: 75
• 75 ÷ 100 = 0.75

4. Decimal to Fraction

This one is a little trickier, but still easy!

• Step 1: Write the decimal as a fraction with a denominator of 1 (e.g., 0.75 becomes 0.75/1).
• Step 2: Multiply both the numerator and the denominator by a power of 10 (10, 100, 1000, etc.) until the numerator is a whole number. The number of zeros in the power of 10 should match the number of decimal places.
• Step 3: Simplify the fraction.

Example: Convert 0.4 to a fraction.

• 0.4 has one decimal place, so multiply by 10/10: (0.4 × 10) / (1 × 10) = 4/10
• Simplify: 4/10 = 2/5

5. Fraction to Percent

To change a fraction to a percent, first convert the fraction to a decimal (divide numerator by denominator), and then convert the decimal to a percent (multiply by 100 and add %).

Example: Convert 1/4 to a percent.

• 1 ÷ 4 = 0.25
• 0.25 × 100 = 25
• Add %: 25%

6. Percent to Fraction

To change a percent to a fraction, write the percent number over 100 and then simplify the fraction.

Example: Convert 60% to a fraction.

• Write over 100: 60/100
• Simplify: 60/100 = 6/10 = 3/5

Practice Makes Perfect!

Now that you have your translation guide, it's time to practice converting between these forms. The more you practice, the more natural it will become!

DFP Practice: Convert These Numbers!

Now that you've learned the different ways to convert between decimals, fractions, and percentages, it's time to test your skills! Use the DFP Conversions Reading as your guide.

Part 1: Convert to Decimal

Instructions: Convert each of the following to a decimal.

1. Fraction: 1/2
   Decimal:
   
   
   

2. Percent: 75%
   Decimal:
   
   
   

3. Fraction: 3/10
   Decimal:
   
   
   

4. Percent: 20%
   Decimal:
   
   
   

Part 2: Convert to Fraction (Simplify if possible)

Instructions: Convert each of the following to a fraction in simplest form.

1. Decimal: 0.25
   Fraction:
   
   
   

2. Percent: 40%
   Fraction:
   
   
   

3. Decimal: 0.8
   Fraction:
   
   
   

4. Percent: 50%
   Fraction:
   
   
   

Part 3: Convert to Percent

Instructions: Convert each of the following to a percentage.

1. Decimal: 0.6
   Percent:
   
   
   

2. Fraction: 3/4
   Percent:
   
   
   

3. Decimal: 0.05
   Percent:
   
   
   

4. Fraction: 1/5
   Percent:
   
   
   

Part 4: Real-World Applications & Critical Thinking

Instructions: Answer the following questions in your own words, using complete sentences.

1. You're at a store, and a shirt is "1/4 off" the original price. Another store offers the same shirt for "25% off". Which store has a better deal? Explain your reasoning using conversions.
   
   
   
   
   
   
   
   
   
   
   
   

2. Why is it important for someone managing their money (e.g., budgeting, looking at interest rates) to understand how to convert between fractions, decimals, and percentages?
   
   
   
   
   
   
   
   
   
   
   
   

3. Imagine you are explaining to a younger student why 0.5 is the same as 50% and 1/2. What simple explanation would you use?
   
   
   
   
   
   
   
   
   
   
   
   

DFP Practice Answer Key

Here are the solutions and explanations for the DFP Practice Worksheet. Review your answers carefully to understand the conversion methods!

Part 1: Convert to Decimal

1. Fraction: 1/2
   Decimal: 0.5

   • Thought Process: Divide 1 by 2 (1 ÷ 2 = 0.5).

2. Percent: 75%
   Decimal: 0.75

   • Thought Process: Remove the % sign and divide by 100 (75 ÷ 100 = 0.75), or move the decimal two places to the left.

3. Fraction: 3/10
   Decimal: 0.3

   • Thought Process: Divide 3 by 10 (3 ÷ 10 = 0.3).

4. Percent: 20%
   Decimal: 0.2

   • Thought Process: Remove the % sign and divide by 100 (20 ÷ 100 = 0.2), or move the decimal two places to the left.

Part 2: Convert to Fraction (Simplify if possible)

1. Decimal: 0.25
   Fraction: 1/4

   • Thought Process: 0.25 is "25 hundredths," so write as 25/100. Simplify by dividing both by 25 (25÷25=1, 100÷25=4).

2. Percent: 40%
   Fraction: 2/5

   • Thought Process: Write the percent over 100 (40/100). Simplify by dividing both by 20 (40÷20=2, 100÷20=5).

3. Decimal: 0.8
   Fraction: 4/5

   • Thought Process: 0.8 is "8 tenths," so write as 8/10. Simplify by dividing both by 2 (8÷2=4, 10÷2=5).

4. Percent: 50%
   Fraction: 1/2

   • Thought Process: Write the percent over 100 (50/100). Simplify by dividing both by 50 (50÷50=1, 100÷50=2).

Part 3: Convert to Percent

1. Decimal: 0.6
   Percent: 60%

   • Thought Process: Multiply the decimal by 100 and add % (0.6 × 100 = 60%), or move the decimal two places to the right.

2. Fraction: 3/4
   Percent: 75%

   • Thought Process: First convert to decimal (3 ÷ 4 = 0.75). Then multiply by 100 and add % (0.75 × 100 = 75%).

3. Decimal: 0.05
   Percent: 5%

   • Thought Process: Multiply the decimal by 100 and add % (0.05 × 100 = 5%), or move the decimal two places to the right.

4. Fraction: 1/5
   Percent: 20%

   • Thought Process: First convert to decimal (1 ÷ 5 = 0.2). Then multiply by 100 and add % (0.2 × 100 = 20%).

Part 4: Real-World Applications & Critical Thinking

1. You're at a store, and a shirt is "1/4 off" the original price. Another store offers the same shirt for "25% off". Which store has a better deal? Explain your reasoning using conversions.
   Both stores are offering the same deal! To compare, we can convert 1/4 to a percentage: 1 ÷ 4 = 0.25, and 0.25 × 100 = 25%. So, 1/4 off is equivalent to 25% off. This means you would save the same amount of money at either store.

2. Why is it important for someone managing their money (e.g., budgeting, looking at interest rates) to understand how to convert between fractions, decimals, and percentages?
   Understanding these conversions is crucial for money management because financial information is often presented in all three forms. For example, interest rates are usually percentages (e.g., 5% interest), discounts might be fractions (e.g., 1/3 off), and calculations in spreadsheets are often done with decimals (e.g., 0.05 for 5%). Being able to convert between them allows a person to accurately compare deals, understand their budget, and make informed financial decisions without getting confused by different representations of numbers.

3. Imagine you are explaining to a younger student why 0.5 is the same as 50% and 1/2. What simple explanation would you use?
   I would explain it like this: Imagine you have a delicious chocolate bar. If you cut it exactly in half, you have 1/2 of the bar. If you look at it as a decimal, that half is 0.5 of the whole bar. And if you think about it as how much out of 100 parts you have, you have 50% of the bar. They all mean the exact same amount – half of the chocolate bar! It's just like saying "hello," "hola," or "bonjour"; different words, same friendly greeting!"

Cryptography Basics: The Art of Secret Writing

Have you ever sent a secret message to a friend? Maybe you used a simple code where 'A' became '1', 'B' became '2', and so on. If you have, you've already dabbled in the fascinating world of cryptography!

Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries. In simpler terms, it's about keeping information secret and safe from people who shouldn't see it. This ancient art has been used for thousands of years by generals, spies, and even lovers to protect their messages.

Why is Cryptography Important?

In today's digital world, cryptography is more important than ever. Think about it: every time you send a text, buy something online, or log into your social media, cryptography is working behind the scenes to protect your personal information. Without it, your passwords, bank details, and private conversations would be open for anyone to see!

Key Terms in Cryptography

To understand cryptography, let's learn some essential vocabulary:

• Plaintext: This is your original, unencrypted message – the information you want to keep secret. For example, "HELLO" is plaintext.
• Ciphertext: This is the encrypted, scrambled version of your message. It's what the message looks like after it's been protected. For example, if "HELLO" becomes "KHOOR" after encryption, "KHOOR" is the ciphertext.
• Encryption: The process of converting plaintext into ciphertext using a specific method or algorithm. It's like putting your secret message into a locked box.
• Decryption: The process of converting ciphertext back into plaintext, so the intended recipient can read it. It's like unlocking the box to read the message.
• Cipher (or Algorithm): This is the set of rules or the specific method used for encryption and decryption. Think of it as the lock and key mechanism. A famous early cipher is the Caesar cipher.
• Key: A secret piece of information (like a password or a number) used with the cipher to encrypt and decrypt messages. Without the correct key, decrypting the message is very difficult.

The Caesar Cipher: A Simple Start

One of the earliest and simplest ciphers is the Caesar cipher, used by Julius Caesar to protect military communications. It's a type of substitution cipher, meaning each letter in the plaintext is replaced by a letter some fixed number of positions down the alphabet.

How it Works:

Let's say the key is 3. This means each letter in the plaintext is shifted three places forward in the alphabet.

• A becomes D
• B becomes E
• C becomes F
• ...and so on.

If we reach the end of the alphabet, we wrap around. So, X becomes A, Y becomes B, and Z becomes C.

Example:

• Plaintext: MATH
• Key: Shift 3
• M (13th letter) + 3 = P (16th letter)
• A (1st letter) + 3 = D (4th letter)
• T (20th letter) + 3 = W (23rd letter)
• H (8th letter) + 3 = K (11th letter)
• Ciphertext: PDKW

To decrypt, you simply shift each letter back by the key number. So, if you receive "PDKW" with a key of 3, you shift each letter back 3 places to get "MATH".

Beyond Caesar

The Caesar cipher is easy to understand, but also easy to break! Modern cryptography uses much more complex ciphers and longer, more intricate keys to ensure security. However, the fundamental concepts of plaintext, ciphertext, encryption, decryption, cipher, and key remain at the heart of all cryptographic systems, from ancient codes to the digital security protecting your life online.

Decoding Challenges: Put Your Skills to the Test!

Now that you've learned the basics of cryptography and the Caesar cipher, it's time to put your decoding skills to the test! Use the information from the Cryptography Basics Reading to solve the following challenges.

Part 1: Caesar Cipher Encryption

Instructions: Encrypt the following messages using the given key (shift amount). Remember to wrap around the alphabet if you go past Z.

1. Plaintext: SECRET
   Key: Shift 3
   Ciphertext:
   
   
   
   
   
   

2. Plaintext: CODE
   Key: Shift 5
   Ciphertext:
   
   
   
   
   
   

3. Plaintext: LEARN
   Key: Shift 1
   Ciphertext:
   
   
   
   
   
   

Part 2: Caesar Cipher Decryption

Instructions: Decrypt the following ciphertexts using the given key (shift amount). Shift backwards to reveal the original message.

1. Ciphertext: WKHVD
   Key: Shift 3
   Plaintext:
   
   
   
   
   
   

2. Ciphertext: FQNW
   Key: Shift 2
   Plaintext:
   
   
   
   
   
   

3. Ciphertext: QEBEEV
   Key: Shift 4
   Plaintext:
   
   
   
   
   
   

Part 3: Critical Thinking

Instructions: Answer the following questions in your own words.

1. Why is the Caesar cipher considered a weak form of encryption in modern times?
   
   
   
   
   
   
   
   
   
   
   
   

2. What is the main difference between plaintext and ciphertext?
   
   
   
   
   
   
   
   
   
   
   
   

3. Imagine you intercept a message encrypted with a Caesar cipher, but you don't know the key. How could you try to figure out the original message?
   
   
   
   
   
   
   
   
   
   
   
   

4. Considering college-level studies or future careers, in what modern scenarios might the principles of cryptography (even simple ones) still be relevant, and what are the ethical considerations?
   
   
   
   
   
   
   
   
   
   
   
   

5. How does the increasing complexity of modern ciphers (beyond what was discussed) relate to advancements in computing power and the need for greater data security?
   
   
   
   
   
   
   
   
   
   
   
   

Decoding Challenges Answer Key

Here are the solutions and explanations for the Decoding Challenges Worksheet. Review your answers carefully to understand the concepts!

Part 1: Caesar Cipher Encryption

1. Plaintext: SECRET
   Key: Shift 3

   • S (19) + 3 = V (22)
   • E (5) + 3 = H (8)
   • C (3) + 3 = F (6)
   • R (18) + 3 = U (21)
   • E (5) + 3 = H (8)
   • T (20) + 3 = W (23)
     Ciphertext: VFHUHW

2. Plaintext: CODE
   Key: Shift 5

   • C (3) + 5 = H (8)
   • O (15) + 5 = T (20)
   • D (4) + 5 = I (9)
   • E (5) + 5 = J (10)
     Ciphertext: HTIJ

3. Plaintext: LEARN
   Key: Shift 1

   • L (12) + 1 = M (13)
   • E (5) + 1 = F (6)
   • A (1) + 1 = B (2)
   • R (18) + 1 = S (19)
   • N (14) + 1 = O (15)
     Ciphertext: MFBSO

Part 2: Caesar Cipher Decryption

1. Ciphertext: WKHVD
   Key: Shift 3 (shift backward)

   • W (23) - 3 = T (20)
   • K (11) - 3 = H (8)
   • H (8) - 3 = E (5)
   • V (22) - 3 = S (19)
   • D (4) - 3 = A (1)
     Plaintext: THESA

2. Ciphertext: FQNW
   Key: Shift 2 (shift backward)

   • F (6) - 2 = D (4)
   • Q (17) - 2 = O (15)
   • N (14) - 2 = L (12)
   • W (23) - 2 = U (21)
     Plaintext: DOLU

3. Ciphertext: QEBEEV
   Key: Shift 4 (shift backward)

   • Q (17) - 4 = M (13)
   • E (5) - 4 = A (1)
   • B (2) - 4 = X (24, wraps around: 2 - 4 = -2, -2 + 26 = 24)
   • E (5) - 4 = A (1)
   • E (5) - 4 = A (1)
   • V (22) - 4 = R (18)
     Plaintext: MAXAAR

Part 3: Critical Thinking

1. Why is the Caesar cipher considered a weak form of encryption in modern times?
   The Caesar cipher is weak because there are only 25 possible keys (shifts). A determined attacker can easily try all 25 possible shifts in a short amount of time, a process called a "brute-force attack," to decrypt the message without knowing the key. Modern encryption methods use much longer keys and more complex algorithms that make brute-force attacks computationally infeasible.

2. What is the main difference between plaintext and ciphertext?
   Plaintext is the original, readable message before any encryption is applied. It's the information you want to protect. Ciphertext is the encrypted, scrambled version of the message. It's unreadable without the correct decryption key and algorithm, serving as the protected form of the plaintext.

3. Imagine you intercept a message encrypted with a Caesar cipher, but you don't know the key. How could you try to figure out the original message?
   You could try a brute-force attack. Since there are only 25 possible shift keys, you can systematically try each possible shift (from 1 to 25) on the ciphertext until you find a readable message. This method would quickly reveal the original plaintext and the key used for encryption.

4. Considering college-level studies or future careers, in what modern scenarios might the principles of cryptography (even simple ones) still be relevant, and what are the ethical considerations?
   Even simple cryptographic principles, like the idea of a

Cryptography Basics: The Art of Secret Writing

Have you ever sent a secret message to a friend? Maybe you used a simple code where 'A' became '1', 'B' became '2', and so on. If you have, you've already dabbled in the fascinating world of cryptography!

Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries. In simpler terms, it's about keeping information secret and safe from people who shouldn't see it. This ancient art has been used for thousands of years by generals, spies, and even lovers to protect their messages.

Why is Cryptography Important?

In today's digital world, cryptography is more important than ever. Think about it: every time you send a text, buy something online, or log into your social media, cryptography is working behind the scenes to protect your personal information. Without it, your passwords, bank details, and private conversations would be open for anyone to see!

Key Terms in Cryptography

To understand cryptography, let's learn some essential vocabulary:

• Plaintext: This is your original, unencrypted message – the information you want to keep secret. For example,