Year 5 Decimals and percentages: check-up and SATs-style questions

Key words. A decimal uses a point (the decimal point) to show parts of a whole. The places after the point are tenths (one of ten equal parts), hundredths (one of a hundred equal parts) and thousandths (one of a thousand equal parts).

Method. Write the digits as the top number and the place value as the bottom number. Then simplify.

1. Read the digits after the point. The last digit is in the hundredths column, so the bottom number is 100.
   0.48: the digits after the point are 48
2. Write the digits over the place value. The top number is the digits. The bottom number is the place value of the last digit.
   ⁴⁸⁄₁₀₀
3. Simplify the fraction. The highest common factor (HCF) of 48 and 100 is 4. Divide the top and the bottom by 4.
   ⁴⁸⁄₁₀₀ = ¹²⁄₂₅
4. Write the answer. This is the answer.
   ¹²⁄₂₅

Check. 48 ÷ 100 = 0.48, so 0.48 is right. ✓

Watch out. Count the decimal places: 2 places means a bottom number of 100. Do not forget the zeros.

Key words. A decimal uses a point (the decimal point) to show parts of a whole. The places after the point are tenths (one of ten equal parts), hundredths (one of a hundred equal parts) and thousandths (one of a thousand equal parts).

Method. Write the digits as the top number and the place value as the bottom number. Then simplify.

1. Read the digits after the point. The last digit is in the thousandths column, so the bottom number is 1,000.
   2.075: the digits after the point are 075
2. Write the digits over the place value. The top number is the digits. The bottom number is the place value of the last digit.
   2 and ⁷⁵⁄₁₀₀₀
3. Simplify the fraction. The highest common factor (HCF) of 75 and 1,000 is 25. Divide the top and the bottom by 25.
   ⁷⁵⁄₁₀₀₀ = ³⁄₄₀
4. Write the answer. Read the whole number and the fraction together.
   2 ³⁄₄₀

Check. 75 ÷ 1,000 = 0.075, so 2.075 is right. ✓

Watch out. Count the decimal places: 3 places means a bottom number of 1,000. Do not forget the zeros.

Key words. A decimal uses a point (the decimal point) to show parts of a whole. The places after the point are tenths (one of ten equal parts), hundredths (one of a hundred equal parts) and thousandths (one of a thousand equal parts).

Method. Make the bottom number 10, 100 or 1,000. Then write the top number with the decimal point in the right place.

1. Make the bottom number 10, 100 or 1,000. 20 × 5 = 100, so multiply the top and the bottom by 5.
   ⁹⁄₂₀ = ⁴⁵⁄₁₀₀
2. Write the decimal point. The bottom number has 2 zeros, so the last digit is in the hundredths column. Fill the empty places with zeros.
   0.45
3. Write the answer.
   0.45

Check. 9 ÷ 20 = 0.45. ✓

Watch out. The bottom number shows where the last digit goes: 100 means hundredths. Write 0.07 for 7 hundredths, not 0.7.

Key words. Place value tells you what each digit is worth. We compare numbers by starting at the left and finding the first place where the digits are different. > means “is greater than”, and < means “is less than”.

Method. Line up the place values, then compare from the left.

1. Write both numbers with the same number of decimal places. Zeros at the end do not change a number. They make the columns line up.
   0.407 = 0.407
   0.41 = 0.410
2. Compare the columns from the left. The first column where the digits are different decides the answer.
   ones: 0 and 0, the same.
   tenths: 4 and 4, the same.
   hundredths: 0 and 1, so 0.41 is bigger.
3. Write the answer. The symbol points to the smaller number.
   0.407 < 0.41

Check. Read it the other way round: 0.407 < 0.41. ✓

Watch out. Do not compare the digits after the point as if they were whole numbers. 0.6 is 0.60, which is bigger than 0.58.

Key words. Place value tells you what each digit is worth. We compare numbers by starting at the left and finding the first place where the digits are different. > means “is greater than”, and < means “is less than”.

Method. Write the numbers with the same number of decimal places, then compare them column by column from the left.

1. Write every number with the same number of decimal places. Adding zeros at the end does not change a decimal. It lines up the place values.
   3.5 = 3.500
   3.45 = 3.450
   3.054 = 3.054
   3.405 = 3.405
2. Put them in order, smallest to largest. Start with the whole numbers. If they are the same, compare the tenths, then the hundredths, then the thousandths.
   3.054 < 3.405 < 3.450 < 3.500
3. Write the answer with the original numbers.
   3.054, 3.405, 3.45, 3.5

Check. Read the list back. Each number is < the next one. ✓

Watch out. Do not sort by the number of digits. 0.45 and 0.405 look like 45 and 405, but 0.45 = 0.450 is the bigger number.

Key words. To round a number is to change it to the nearest value in a given place, such as the nearest whole number or one decimal place. The halfway number is exactly between two numbers. A digit of 5 or more rounds up; a digit of 4 or less rounds down.

Method. Find the two numbers it lies between, find the halfway number, then see which one it is nearer to.

1. Find the two numbers it is between. We round to one decimal place. These are the two numbers it could round to.
   5.65 is between 5.6 and 5.7
2. Find the halfway number. The halfway number is exactly between the two. A number at or above halfway rounds up.
   Halfway = 5.65
3. Look at the digit after the place you are rounding to. The digit is 5. 5 or more: round up.
   5.65 is exactly on halfway, so round up.
4. Write the answer. Keep the place you rounded to, and drop the digits after it.
   5.7

Check. The difference is 0.05, which is at most half a step (0.05). ✓

Watch out. Look at ONE digit only. Do not round twice: 3.449 is 3.4 to one decimal place, not 3.5.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

1. Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.

     6.45 
   + 3.675
   -------

2. Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.

     6.450
   + 3.675
   -------

3. Add column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.

      6.450
   +  3.675
   --------
     10.125

4. Write the decimal point in the answer. Bring the point straight down, under the others.
   10.125

Check. Check with the opposite operation: 10.125 − 3.675 = 6.45. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

1. Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.

     8.  
   − 2.46
   ------

2. Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.

     8.00
   − 2.46
   ------

3. Subtract column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.

     8.00
   − 2.46
   ------
     5.54

4. Write the decimal point in the answer. Bring the point straight down, under the others.
   5.54

Check. Check with the opposite operation: 5.54 + 2.46 = 8. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Change the decimal to hundredths. The number of hundredths is the percentage.

1. Write the decimal as hundredths. Hundredths are the same as per cent: 1 hundredth = 1%.
   0.65 = ⁶⁵⁄₁₀₀
2. Read the number of hundredths as the percentage. Multiplying by 100 moves the digits two places to the left.
   0.65 × 100 = 65
3. Write the answer.
   65%

Check. 65% = ⁶⁵⁄₁₀₀ = 0.65. ✓

Watch out. Multiplying by 100 moves the digits two places to the LEFT. Dividing by 100 moves them two places to the right.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Scale the fraction so that the bottom number is 100. The top number is then the percentage.

1. Make the bottom number 100. Per cent means out of 100. 5 × 20 = 100, so multiply the top and the bottom by 20.
   ³⁄₅ = ⁶⁰⁄₁₀₀
2. Read the top number as the percentage. The top number, out of 100, is the percentage.
   ⁶⁰⁄₁₀₀ = 60%
3. Write the answer.
   60%

Check. 60 ÷ 100 = 0.6, and 3 ÷ 5 = 0.6. Both are the same decimal. ✓

Watch out. Multiply the top AND the bottom by the same number. Do not add to them.

Key words. A decimal uses a point (the decimal point) to show parts of a whole. The places after the point are tenths (one of ten equal parts), hundredths (one of a hundred equal parts) and thousandths (one of a thousand equal parts).

Method. Find the column of the 7, then work out its value.

1. Look at each digit in its column. The places after the point are tenths, hundredths and thousandths.
   3 ones, 4 tenths, 7 hundredths, 8 thousandths
2. Find the column of the 7. The 7 is in the second place after the point, so it is in the hundredths column.
   hundredths column
3. Work out its value. 7 hundredths is 7 × 0.01.
   7 × 0.01 = 0.07

Check. 0.07 is 7 out of 100. ✓

Watch out. The 7 is not 7 tenths. Count the places after the point.

Key words. A decimal uses a point (the decimal point) to show parts of a whole. The places after the point are tenths (one of ten equal parts), hundredths (one of a hundred equal parts) and thousandths (one of a thousand equal parts).

Method. Write the digits as the top number and the place value as the bottom number. Then simplify.

1. Read the digits after the point. The last digit is in the thousandths column, so the bottom number is 1,000.
   0.085: the digits after the point are 085
2. Write the digits over the place value. The top number is the digits. The bottom number is the place value of the last digit.
   ⁸⁵⁄₁₀₀₀
3. Simplify the fraction. The highest common factor (HCF) of 85 and 1,000 is 5. Divide the top and the bottom by 5.
   ⁸⁵⁄₁₀₀₀ = ¹⁷⁄₂₀₀
4. Write the answer. This is the answer.
   ¹⁷⁄₂₀₀

Check. 85 ÷ 1,000 = 0.085, so 0.085 is right. ✓

Watch out. Count the decimal places: 3 places means a bottom number of 1,000. Do not forget the zeros.

Key words. Place value tells you what each digit is worth. We compare numbers by starting at the left and finding the first place where the digits are different. > means “is greater than”, and < means “is less than”.

Method. Line up the place values, then compare from the left.

1. Write both numbers with the same number of decimal places. Zeros at the end do not change a number. They make the columns line up.
   0.45 = 0.450
   0.405 = 0.405
2. Compare the columns from the left. The first column where the digits are different decides the answer.
   ones: 0 and 0, the same.
   tenths: 4 and 4, the same.
   hundredths: 5 and 0, so 0.45 is bigger.
3. Write the answer. The symbol points to the smaller number.
   0.45 > 0.405

Check. Read it the other way round: 0.405 < 0.45. ✓

Watch out. Do not compare the digits after the point as if they were whole numbers. 0.6 is 0.60, which is bigger than 0.58.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

1. Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.

     4.6  
   + 7.085
   -------

2. Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.

     4.600
   + 7.085
   -------

3. Add column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.

      4.600
   +  7.085
   --------
     11.685

4. Write the decimal point in the answer. Bring the point straight down, under the others.
   11.685

Check. Check with the opposite operation: 11.685 − 7.085 = 4.6. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

1. Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.

     9.   
   − 3.275
   -------

2. Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.

     9.000
   − 3.275
   -------

3. Subtract column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.

     9.000
   − 3.275
   -------
     5.725

4. Write the decimal point in the answer. Bring the point straight down, under the others.
   5.725

Check. Check with the opposite operation: 5.725 + 3.275 = 9. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

1. Take away the first amount. The bottle had 1.5 litres. Sam poured out 0.375 litres first, so subtract it.
   1.5 litres − 0.375 litres = 1.125 litres
2. Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.

     1.125
   − 0.2  
   -------

3. Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.

     1.125
   − 0.200
   -------

4. Subtract column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.

     1.125
   − 0.200
   -------
     0.925

5. Write the decimal point in the answer. Bring the point straight down, under the others.
   0.925 litres

Check. Check with the opposite operation: 0.925 + 0.2 = 1.125. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Key words. Equivalent means the same value written in a different way. A fraction, a decimal and a percentage can all be equivalent to each other.

Method. Change each one to a decimal, then tick the ones that equal 0.4.

1. Change each one to a decimal. Make the bottom number 10, 100 or 1,000, or use the percentage.

   Number	Decimal	Equal to 0.4?
   2⁄5	2/5 = 0.4	yes
   4⁄10	4/10 = 0.4	yes
   40%	40% = 0.4	yes
   4⁄100	4/100 = 0.04	no
   0.04	0.04	no

2. Tick the ones that are equal to 0.4. Three of them have the value 0.4.
   2/5, 4/10 and 40%
3. Write the answer. Check that you have not ticked the ones that are not 0.4.
   2/5, 4/10 and 40%

Check. 2/5, 4/10 and 40% are all 0.4. ✓

Watch out. 4/100 is 0.04, not 0.4. Check the number of decimal places.

Key words. To round a number is to change it to the nearest value in a given place, such as the nearest whole number or one decimal place. The halfway number is exactly between two numbers. A digit of 5 or more rounds up; a digit of 4 or less rounds down.

Method. Change every number to a percentage, then put them in order from the smallest.

1. Change each number to a percentage. Multiply a decimal by 100, or make the fraction out of 100.

   Number	As a percentage
   0.4	40%
   45%	45%
   0.405	40.5%
   3⁄10	30%

2. Put them in order. Compare the percentages from the smallest.
   30% < 40% < 40.5% < 45%
3. Write the answer in the original form. The numbers are the same, so write them as they were given.
   ³⁄₁₀, 0.4, 0.405, 45%

Check. 0.405 = 40.5%, which is between 40% and 45%. ✓

Watch out. Compare with the same unit. The number 0.405 is 40.5%, not 4.05%.