Fiche de révision : Mastering Fractions, Decimals, and Ratios

📋 Course Outline

1. Fractions in Decimals
2. Converting Fractions
3. Fraction of a Quantity
4. Percentages Calculation
5. Percentage Increase/Decrease
6. Interest and GST
7. Ratio Simplification
8. Ratio Conversion and Splitting
9. Rate Calculations
10. Standard Form Numbers

📖 1. Fractions in Decimals

🔑 Key Concepts & Definitions

• Converting decimal amounts spent into fractions using calculator (SD button):
  This process involves entering the decimal amount as the numerator and the total amount as the denominator, then pressing the SD (simplify) button on the calculator to automatically reduce the fraction to its simplest form. For example, to convert $24.80 spent out of$40, input 24.80/40, then press SD to get 31/50.

• Subtracting fractions from a whole to find remaining fraction:
  To determine the part of a whole that remains after subtracting fractions, convert the fractions to a common denominator or decimal form, then subtract from 1 (the whole). For example, if you have 1/8 and 2/5, convert and subtract to find the remaining fraction.

• Finding the fraction of a quantity by dividing by the denominator and multiplying by the numerator:
  To find a specific fraction of a quantity, divide the total by the denominator, then multiply the result by the numerator. For example, to find 3/7 of 560kg, divide 560 by 7, then multiply by 3, resulting in 240kg.

• Understanding not to leave decimals in fraction form:
  When working with fractions, always convert decimals into simplified fractions using calculator functions (like SD) or by expressing them as ratios, rather than leaving them as decimal numbers, to ensure clarity and accuracy in calculations.

📝 Essential Points

• Use calculator SD button to simplify fractions derived from decimal amounts, ensuring the fraction is in its lowest terms.
• When subtracting fractions from a whole, convert all fractions to a common denominator or decimal form to accurately determine the remaining part.
• To find a fraction of a quantity, divide the total by the denominator and multiply by the numerator; this method is essential for proportional calculations.
• Always avoid leaving decimals in fractional form; convert to a simplified fraction for precise and clear communication, especially in exams.
• These techniques streamline calculations involving fractions and decimals, reducing errors and improving efficiency.

💡 Key Takeaway

Mastering the conversion of decimal amounts into simplified fractions using calculator functions, and understanding how to manipulate fractions through subtraction and multiplication, is crucial for accurate proportional and fractional calculations.

📖 2. Converting Fractions

🔑 Key Concepts & Definitions

• Writing fractions as decimals using calculator and rounding: To convert a fraction to a decimal, input the numerator, press the division key, then the denominator, and press enter. Use the calculator's SD (significant digits) button to round the decimal to the desired number of decimal places, especially when working with money or measurements (see examples for money and measurements).

• Converting fractions to percentages by multiplying decimal by 100: After converting a fraction to a decimal, multiply the decimal by 100 to express it as a percentage. This process involves first converting the fraction to a decimal (see above), then performing the multiplication, often with rounding if necessary.

📝 Essential Points

• When converting fractions to decimals, do not leave the decimal in fraction form; always use the calculator to obtain a decimal value. For example, input 24.80 over 40, then press SD to get the simplified fraction as 31/50.

• To convert a decimal to a percentage, multiply the decimal by 100. For example, a decimal of 0.52 becomes 52% after multiplication.

• Rounding is crucial, especially in financial contexts (e.g., money calculations), where adding a zero after the decimal point ensures accuracy (e.g., $106.40 from 28%).

• When converting fractions like 1/8 or 2/5, first convert to decimal, then multiply by 100 to get the percentage. For example, 1/8 as a decimal is 0.125, which becomes 12.5%.

• For fractions involving measurements or weights, convert to decimal first, then to percentage if needed, ensuring units are consistent before calculations.

💡 Key Takeaway

Converting fractions to decimals using calculator and rounding simplifies the process of expressing fractions in decimal form, which can then be easily transformed into percentages by multiplying by 100. This method ensures accuracy and clarity in calculations involving parts of a whole.

📖 3. Fraction of a Quantity

🔑 Key Concepts & Definitions

• Finding a fraction of a quantity: To find a fraction of a quantity, divide the total amount by the denominator of the fraction, then multiply the result by the numerator.
  Example: To find 3/7 of 560 kg, divide 560 by 7, then multiply by 3:
  $560 \div 7 \times 3 = 240$ kg.

• Subtracting fractions from 1 to find the remaining fraction: To determine the part of a whole not taken by certain fractions, subtract those fractions from 1 (the whole).
  Example: If 1/8 and 2/5 are parts of a whole, the remaining fraction (bike, for example) is:
  $1 - \frac{1}{8} - \frac{2}{5} = \frac{11}{40}$.

• Expressing a fraction as a decimal or percentage: Convert the fraction into a decimal by dividing numerator by denominator, then multiply by 100 for percentage.
  Example: $\frac{21}{51}$ as a percentage:
  $21 \div 51 \approx 0.4118$, then $0.4118 \times 100 \approx 41.18\%$.

📝 Essential Points

• To find a specific fraction of a quantity, always divide the total by the denominator and then multiply by the numerator, as demonstrated in the examples.
• When subtracting fractions from 1, ensure all fractions are expressed with a common denominator or converted to decimals for accuracy.
• Converting fractions to percentages involves dividing the numerator by the denominator to get a decimal, then multiplying by 100.
• For complex calculations, such as finding remaining fractions, it’s often easier to work with common denominators or convert to decimals before subtraction.
• Always leave fractions in their simplest form unless specified otherwise, especially when working with ratios or percentages.

💡 Key Takeaway

To find a fraction of a quantity, divide the total by the denominator and multiply by the numerator; to find the remaining part, subtract known fractions from 1. Converting fractions to decimals or percentages helps in easier interpretation and comparison.

📖 4. Percentages Calculation

🔑 Key Concepts & Definitions

• Calculating percentage of an amount: To find a specific percentage of an amount, divide the total amount by 100, then multiply the result by the percentage.
  Example: To find 28% of $380, compute$ 380 \div 100 = 3.8 $, then$ 3.8 \times 28 = $106.40 $.
  Note: Always add a zero when working with money to maintain accuracy.

• Converting fraction to decimal then to percentage: Convert a fraction to a decimal by dividing numerator by denominator, then multiply the decimal by 100 to get the percentage.
  Example: Convert $\frac{21}{51}$ to decimal, then to percentage by multiplying by 100.

• Calculating percentage increase or decrease: Find the difference between the two amounts, divide that difference by the original amount, then multiply by 100 to get the percentage change.
  Example: Increase from 340 to 359: $\frac{19}{340} \times 100 = 5.6\%$.
  For decrease, use the higher number as the denominator.

• Adding a zero when working with money in percentage calculations: When calculating percentages involving money, always append a zero to the percentage or decimal to ensure the calculation reflects monetary values correctly.
  Example: To find 28% of $380, use$ 3.8 \times 28 $instead of$ 3.8 \times 2.8 $.

📝 Essential Points

• To find a percentage of an amount, divide the total by 100, then multiply by the percentage.
• Convert fractions to decimals by dividing numerator by denominator, then multiply by 100 to get the percentage.
• When calculating percentage increase or decrease, always use the original amount as the denominator for accuracy.
• Always add a zero when working with money to avoid miscalculations in percentage problems involving currency.
• For percentage change, the formula is: $\frac{\text{difference}}{\text{original}} \times 100$.
• To convert a fraction to a percentage, first convert to decimal, then multiply by 100.

💡 Key Takeaway

Mastering percentage calculations involves converting fractions to decimals, applying the correct formulas for increase or decrease, and always adjusting for money by adding zeros to maintain precision.

📖 5. Percentage Increase/Decrease

🔑 Key Concepts & Definitions

Calculating percentage increase: To find the percentage increase, multiply the original amount by (1 + decimal percentage).
Example: Increasing $100 by 20$120.

Calculating percentage decrease: To find the percentage decrease, multiply the original amount by (1 - decimal percentage).
Example: Decreasing $200 by 15$170.

Adjusting multiplier for over 100% increase: When the percentage increase exceeds 100%, convert it to a decimal by dividing by 100 and then add 1. For example, 148% becomes 1 + 1.48 = 2.48.
Example: Increasing $50 by 148$124.

Finding percentage increase or decrease: Subtract the original amount from the new amount, then divide the difference by the original amount and multiply by 100.
Example: From $340 to$359: (359 - 340) / 340 × 100 ≈ 5.6% increase.

📝 Essential Points

• To calculate percentage increase, multiply the original amount by (1 + decimal percentage). This method simplifies calculations, especially with money, where always add a zero after the decimal (e.g., 28% as 0.28).
• For percentage decrease, multiply by (1 - decimal percentage).
• When percentage increase exceeds 100%, adjust the multiplier by adding the percentage as a decimal to 1 (e.g., 148% as 2.48).
• To find the percentage change, always compute the difference between the two amounts, then divide by the original amount, and multiply by 100 to convert to a percentage.
• For decreases, use the higher number as the denominator; for increases, use the lower number.

💡 Key Takeaway

Calculating percentage increase or decrease involves multiplying by a decimal form of the percentage, adjusting for over 100% increases, and using the difference over the original amount times 100 to find the exact percentage change.

📖 6. Interest and GST

🔑 Key Concepts & Definitions

GSTE (GST Excluded):
The price or amount that does not include Goods and Services Tax. It is the original amount before GST is added.
(Understanding GSTE means GST not included)

GSTI (GST Included):
The total price or amount that already includes Goods and Services Tax. It is the original amount plus GST.
(Understanding GSTI means GST is included)

Calculating GST Included Amount:
To find the total amount including GST, multiply the original amount by 1.15.
(Example: $100 x 1.15 =$115)

Calculating GST Excluded Amount:
To find the amount before GST was added, divide the total amount by 1.15.
(Example: $115 ÷ 1.15 =$100)

📝 Essential Points

• GST (Goods and Services Tax) is always 15%.
• To find the GST-inclusive price from a GST-exclusive amount, multiply by 1.15.
• To determine the GST-exclusive amount from a GST-inclusive total, divide by 1.15.
• The terms GSTE and GSTI specify whether GST is excluded or included in the given amount.
• These calculations are crucial for financial transactions, invoicing, and understanding the true cost or price of goods/services.

💡 Key Takeaway

To switch between GST included and excluded amounts, multiply by 1.15 for GST-included totals, and divide by 1.15 for GST-exclusive amounts, ensuring accurate financial calculations.

📖 7. Ratio Simplification

🔑 Key Concepts & Definitions

• Simplifying ratios by converting ratio to fraction and simplifying: To simplify a ratio, convert it into a fraction by placing the first number over the second, then reduce the fraction to its lowest terms by dividing numerator and denominator by their highest common factor (HCF). For example, ratio 8:12 becomes 8/12, which simplifies to 2/3, resulting in the ratio 2:3.

• Simplifying ratios with more than two numbers by dividing all by highest common factor: When a ratio involves multiple numbers, identify the HCF of all the numbers. Divide each number by this HCF to obtain the simplified ratio. For example, 6:12:18 has an HCF of 6; dividing each by 6 yields 1:2:3.

• Ensuring units are the same type before simplifying ratios involving measurements: Before simplifying ratios that involve measurements, convert all units to the same type. For example, convert 2cm:1m into 2cm:100cm to ensure both are in centimeters, making the ratio 2:100, which simplifies to 1:50.

📝 Essential Points

• To simplify a ratio, always convert it into a fraction and reduce it to lowest terms using the HCF. This process ensures the ratio is in its simplest form, making comparisons and calculations easier.
• For ratios with multiple terms, find the HCF of all involved numbers and divide each term by this factor. This maintains the proportional relationship while reducing the numbers.
• When measurements are involved, units must be consistent. Convert different units to a common measurement before simplifying to avoid incorrect ratios.
• Conversion rates between units (e.g., 10mm = 1cm, 100cm = 1m) are essential for ensuring units are compatible before simplifying ratios involving measurements.

💡 Key Takeaway

Simplifying ratios involves converting to fractions, reducing by the highest common factor, and ensuring units are consistent, which makes ratios easier to interpret and work with accurately.

📖 8. Ratio Conversion and Splitting

🔑 Key Concepts & Definitions

• Splitting a quantity into parts according to a ratio: To divide a total amount into specified parts based on a ratio, first find the sum of the ratio parts, then divide the total by this sum, and multiply the result by each ratio part.
  Example: For $72 split in ratio 2:7, total parts = 2 + 7 = 9. Divide$72 by 9 to get 8, then multiply by 2 and 7 for each part: $16 and$56.

• Finding hidden ratio values: When a known quantity corresponds to one part of a ratio, divide the known amount by that ratio part to find the value of one unit, then multiply by other ratio parts to find unknown quantities.
  Example: If 30 units of flour correspond to 5 parts, divide 30 by 5 to get 6, then multiply by the other ratio part (e.g., sugar ratio 1) to find sugar amount.

• Converting ratios with different units to the same units before calculations: Ensure all measurements involved in ratios are expressed in the same units before performing calculations. This may involve converting units (e.g., cm to m, mg to g).
  Example: Convert 2cm:1m to 2cm:100cm before comparing or splitting.

📝 Essential Points

• To split a total into ratio parts, divide the total by the sum of the ratio parts, then multiply by each individual part. This method simplifies the process and ensures proportional distribution.
• When finding hidden values, divide the known quantity by the ratio part it corresponds to, then multiply the result by other ratio parts to find unknown quantities.
• Always convert units to the same measurement system before performing ratio calculations involving different units to maintain accuracy and consistency.
• For ratios involving weights and distances, convert units (e.g., cm to m) to ensure ratios are comparable.
• Use calculator functions (e.g., fraction or division) to simplify ratios and avoid leaving decimal approximations when possible, especially in money or measurement contexts.
• When simplifying ratios with more than two numbers, divide all parts by their highest common factor to reduce the ratio to its simplest form.

💡 Key Takeaway

Splitting a quantity according to a ratio involves dividing the total by the sum of the ratio parts and multiplying by each part, while finding hidden ratio values requires dividing known quantities by their ratio parts and multiplying to find unknowns. Always convert units to the same system before calculations to ensure accuracy.

📖 9. Rate Calculations

🔑 Key Concepts & Definitions

• Unit Rate: The cost or quantity per single item, calculated by dividing the total quantity by the number of items. For example, liters per chair is found by dividing total liters by the number of chairs. (Source: Example calculations)

• Using Unit Rate to Find Total: To find the total for a different quantity, multiply the unit rate by the new number of items. For example, if one chair requires 1.4 liters, then 37 chairs require $37 \times 1.4 = 51.8$ liters. (Source: Example 1)

• Using Exchange Rates to Convert Currencies: To convert an amount from one currency to another, multiply the amount by the exchange rate. For example, converting 50 NZD to USD at a rate of 0.60: $50 \times 0.60 = 30$ USD. (Source: Example 2)

📝 Essential Points

• To calculate unit rate, divide the total quantity by the number of items, then use this rate to find totals for different quantities by multiplication.
• When converting currencies, always multiply the amount by the exchange rate to get the equivalent in the target currency.
• In rate calculations, maintaining consistent units is crucial, especially when dealing with measurements like weight or distance (see section 8).
• Remember, for unit rates involving money, always add a zero when working with percentages or monetary amounts to ensure accuracy (e.g., 28% of $380 is calculated as$380 \div 100 \times 28 = 106.40$).

💡 Key Takeaway

Rate calculations involve understanding how to find and use unit rates to determine totals for different quantities and to convert currencies accurately by multiplying amounts by exchange rates.

📖 10. Standard Form Numbers

🔑 Key Concepts & Definitions

• Writing numbers in standard form by moving decimal point right for positive powers:
  To express a number in standard form with a positive exponent, move the decimal point to the right until only one non-zero digit remains to the left, then multiply by 10 raised to the number of places moved.
  Example: $4.8 \times 10^3 = 4800$.

• Writing numbers in standard form by moving decimal point left for negative powers and adding zeros:
  For negative exponents, move the decimal point to the left the number of places indicated by the negative power, adding zeros if necessary, to convert to standard form.
  Example: $1.4 \times 10^{-6} = 0.0000014$.

• Converting standard form back to normal number:
  To revert from standard form, move the decimal point right for positive exponents or left for negative exponents, adjusting with zeros as needed.
  Example: $3.2 \times 10^4 = 32000$.

• Writing normal numbers in standard form by placing decimal after first digit and counting digits moved:
  To write a normal number in standard form, position the decimal after the first digit, then count how many places the decimal has moved, which becomes the exponent of 10.
  Example: 12000 becomes $1.2 \times 10^4$.

📝 Essential Points

• When converting to standard form, always ensure only one non-zero digit precedes the decimal point.
• Moving the decimal point to the right (positive powers) increases the number's magnitude; moving it left (negative powers) decreases it.
• Adding zeros is necessary when moving the decimal point left for very small numbers, ensuring the number's value remains unchanged.
• To convert from standard form to a normal number, reverse the process: move the decimal point in the opposite direction, adding zeros if needed.
• Counting the number of places moved determines the exponent in standard form, which indicates the scale of the number.

💡 Key Takeaway

Standard form simplifies very large or very small numbers by expressing them as a product of a number between 1 and 10 and a power of 10, making calculations and comparisons more manageable.

📊 Synthesis Tables

⚠️ Common Pitfalls & Confusions

1. Leaving decimals in fractional form instead of converting to simplified fractions using SD or ratios.
2. Forgetting to round decimals appropriately, especially in financial contexts.
3. Confusing numerator and denominator when converting fractions to decimals or percentages.
4. Using incorrect denominators when calculating a fraction of a quantity, leading to wrong results.
5. Mixing up percentage increase and decrease formulas; reversing numerator and denominator.
6. Not simplifying ratios before converting or splitting, resulting in inaccurate division.
7. Forgetting to convert fractions to decimals before multiplying to find percentages.
8. Misapplying the calculator SD function, leading to incorrect simplified fractions.
9. Failing to convert large numbers into standard form, causing calculation errors.
10. Ignoring units (e.g., money, weight) when converting between fractions, decimals, and percentages.

✅ Exam Checklist

• Know how to convert decimal amounts into simplified fractions using calculator SD button.
• Be able to subtract fractions from a whole to find remaining parts.
• Understand how to find a fraction of a quantity by dividing and multiplying.
• Convert fractions to decimals and percentages accurately, rounding as needed.
• Calculate percentages of amounts by dividing total by 100 and multiplying.
• Master percentage increase and decrease formulas, including identifying the correct base.
• Know how to apply interest calculations and GST percentages.
• Simplify ratios by dividing numerator and denominator by their GCD.
• Convert ratios into parts and split quantities proportionally.
• Calculate rates such as speed, density, or unit rates.
• Express large or small numbers in standard form correctly.
• Know SMITH's definition of the invisible hand and its role in market efficiency.
• Be familiar with key authors and their contributions to economic theory.
• Understand the relationship between fractions, decimals, and percentages in various contexts.
• Practice converting between different forms of numbers and ratios efficiently.
• Recognize common pitfalls in fraction and percentage calculations.
• Be able to interpret and manipulate ratios for real-world problems.
• Review all calculator functions related to fractions and standard form.
• Confirm mastery of key concepts through practice questions and exercises.