Section A — Chapters 1–22
Decimals and Percentages
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to understand the place value of digits in decimal numbers.
- ✓By the end of this lesson students will be able to convert between fractions and decimals.
- ✓By the end of this lesson students will be able to round decimal numbers to a specified number of decimal places or significant figures.
- ✓By the end of this lesson students will be able to express a part of a whole as a percentage.
- ✓By the end of this lesson students will be able to calculate percentages of quantities.
Key concepts
Decimals are a way of writing numbers that are not whole numbers. They use a decimal point to separate the whole number part from the fractional part. Each digit after the decimal point represents a decreasing power of ten: the first digit is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), and so on.
Fractions and decimals are different ways of representing parts of a whole. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal as a fraction with a denominator that is a power of 10 (e.g., 0.7 as 7/10, 0.25 as 25/100), then simplify the fraction to its lowest terms.
Rounding a decimal number means approximating it to a certain number of decimal places. To do this, identify the digit in the last decimal place you want to keep. Then, look at the digit immediately to its right. If this digit is 5 or greater, round up the last digit you are keeping (add 1 to it). If it is less than 5, keep the last digit as it is.
Rounding to significant figures involves identifying the most important digits in a number. The first non-zero digit from the left is the first significant figure. To round, count the required number of significant figures from the first non-zero digit, then apply the same rounding rule: if the next digit is 5 or greater, round up; otherwise, keep it the same. For whole numbers, trailing zeros may or may not be significant depending on context, but for decimals, all digits after the first non-zero digit are significant.
A percentage is a way of expressing a number as a fraction of 100. The word 'percent' means 'per hundred' or 'out of 100'. The symbol for percentage is %. To convert a decimal to a percentage, multiply by 100. To convert a fraction to a percentage, first convert it to a decimal, then multiply by 100.
Key facts to remember
- 1The place value of digits after the decimal point are tenths, hundredths, thousandths, etc.
- 2To convert a fraction to a decimal, divide the numerator by the denominator.
- 3To convert a decimal to a fraction, write it over a power of 10 (e.g., 0.3 = 3/10, 0.17 = 17/100) and simplify the fraction.
- 4When rounding, if the digit immediately to the right of the rounding position is 5 or more, round up; otherwise, keep the digit the same.
- 5Percentages mean 'out of 100'. So, 25% is equivalent to 25/100 or 0.25.
- 6To find a percentage of a number, convert the percentage to a decimal (by dividing by 100) and then multiply it by the number.
Worked examples
Example 1
Convert the fraction 5/8 to a decimal and then round the decimal to 2 decimal places.
Answer
0.63
Always identify the digit you are rounding *to* and the digit *after* it to apply the rounding rule correctly.
Example 2
Round the number 47.038 to 3 significant figures.
Answer
47.0
Remember to include the trailing zero if it is significant, as it shows the level of precision.
Example 3
A shop offers a 15% discount on a jacket that originally costs €80. How much is the discount, and what is the new price of the jacket?
Answer
The discount is €12. The new price of the jacket is €68.
Alternatively, if there is a 15% discount, you pay 100% - 15% = 85% of the original price. So, New Price = 0.85 × €80 = €68.
Common mistakes
- ✗Incorrectly identifying place values (e.g., confusing the tenths place with the tens place).
- ✗Rounding down when the next digit is 5 (e.g., rounding 0.45 to 0.4 instead of 0.5).
- ✗Confusing rounding to decimal places with rounding to significant figures, leading to incorrect answers.
- ✗Not simplifying fractions to their lowest terms after converting from decimals.
- ✗Incorrectly converting between percentages and decimals (e.g., multiplying by 100 when it should be dividing, or vice-versa, when finding a percentage of a quantity).
Exam tips
- ★Always read the question carefully to see if you need to round, and to how many decimal places (d.p.) or significant figures (s.f.).
- ★Show all your working steps clearly, especially for conversions between fractions, decimals, and percentages, as this can earn you partial marks.
- ★Use your calculator efficiently for decimal and percentage calculations, but be prepared to show manual steps for simpler conversions or estimations.
- ★Double-check your answers, particularly for rounding, to ensure you've applied the rules correctly.
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