Year 5: Solving number and practical problems
Part of the Year 5 place value to 1,000,000 chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Solving number and practical problems
Now you can put everything together. Problems about big numbers can use place value, ordering, counting in powers of 10, rounding, negative numbers and Roman numerals. Here is a plan that works for every problem.
- Read the problem twice. Underline the numbers and the question.
- Picture it. Draw a number line, a table, a chart or a bar model.
- Choose what to do: count on, compare, round, add or take away.
- Work it out one step at a time.
- Check: does the answer make sense? Use an estimate.
Look at the bar chart. Always read the scale and the labels before you answer. The numbers on the bars are exact, so you can round them, compare them and find differences.
Key words. A clue is a fact that helps you find an unknown number. When a problem has several clues, use them one at a time and cross out the numbers that do not fit.
Key ideas
- Plan: read, picture, choose, work out, check.
- Round the real numbers, not numbers that have already been rounded.
- With clues, start with the clue that gives the most information.
- Use an estimate to check that an answer is sensible.
- Say what your answer means (units, names, years).
Method. Round every number to the nearest 100,000. Then see which ones are the same.
- Round each number. Look at the ten thousands digit each time: 5 or more rounds up, 4 or less rounds down.Park A: 245,000 (digit 4) ≈ 200,000
Park B: 318,500 (digit 1) ≈ 300,000
Park C: 287,000 (digit 8) ≈ 300,000
Park D: 96,200 (digit 9) ≈ 100,000 - Find the numbers that are the same. Park B and Park C both round to 300,000.
- Write the answer. Park B and Park C
Check. The other parks round to different numbers. ✓
Watch out. Round the real numbers, not the rounded ones, each time.
Method. First find the range of numbers that round to 600,000. Then use the cards to make the largest number in that range.
- Find the range for 600,000. To the nearest 100,000, the numbers that round to 600,000 go from 550,000 to 649,999.550,000 to 649,999
- Choose the first digit. The number must be at most 649,999, so the first digit is 6.First digit = 6
- Choose the next digit. The number must still be at most 649,999, so the second digit has to be 4 or less. Take the biggest card that fits.Second digit = 3
- Put the other cards in order, biggest first. This makes the number as big as possible.638,510
Check. 638,510 is between 550,000 and 649,999 and uses every card once. ✓
Watch out. Check your answer rounds to the right number.
Rounded to the nearest 100,000 it is 400,000.
Rounded to the nearest 10,000 it is 350,000.
The thousands digit is 4.
The tens digit and ones digit are the same. The hundreds digit is 2 more than the ones digit.
The digits add up to 26.
What is the number?
Method. Use the rounding clues first to find the range. Then use the digit clues.
- Use the clue about 10,000. To round to 350,000 the number is from 345,000 to 354,999.345,000 to 354,999
- Use the clue about 100,000. To round to 400,000 the number must be at least 350,000. Together with the first clue:350,000 to 354,999. So the number starts 35 and the thousands digit is 0 to 4.
- Use the thousands digit. The thousands digit is 4.354,…
- Use the last three digits. Let the ones digit be o. The tens digit is o and the hundreds digit is o + 2. The digits 3, 5, 4 add to 12, so 12 + (o + 2) + o + o = 26.14 + 3o = 26
3o = 12, so o = 4 - Write the number. Ones 4, tens 4, hundreds 6.354,644
Check. 354,644 ≈ 350,000 (10,000) ✓, ≈ 400,000 (100,000) ✓, thousands 4 ✓, 4 = 4 and 6 = 4 + 2 ✓, 3+5+4+6+4+4 = 26 ✓
Watch out. Check your answer against every clue.
Method. Add 10,000 each month until the total is 300,000 or more.
- Start with the money now. Month 0.Month 0: £235,000
- Add £10,000 each month. Only the ten thousands digit changes.Month 1: £245,000
Month 2: £255,000
Month 3: £265,000
Month 4: £275,000
Month 5: £285,000
Month 6: £295,000
Month 7: £305,000 - Find the first total of 300,000 or more. Month 6 is £295,000, which is not enough. Month 7 is £305,000.7 months
Check. 235,000 + 7 × 10,000 = 305,000, which is at least 300,000. ✓
Watch out. £295,000 is close, but it is less than £300,000, so it is not enough.
Method. Do the moves one at a time. Going up adds. Going down takes away.
- Start position. Below sea level is negative.−24 m
- Swim up 9 m. Going up makes the number bigger.−24 m up 9 m: start below zero and go up 9 m → −15 m
- Dive down 15 m. Going down makes the number smaller.−15 m down 15 m: start at or below zero and go down 15 m more → −30 m
- Write the answer. The question asks how far below sea level, so give the depth.She is at −30 m, which is 30 m below sea level.
Check. Total change = +9 − 15 = −6. −24 − 6 = −30. ✓
Watch out. Take the moves in order and watch which way each one goes.
Key words. Roman numerals are letters used for numbers: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000. A small symbol before a bigger one is taken away; after it is added.
Method. Change the Roman numeral to a year, then round it.
- Change MDCCLXXXIX to a number. M = 1,000, D = 500, CC = 200, LXXX = 80, IX = 9.1,000 + 500 + 200 + 80 + 9 = 1789
- Look at the tens digit. Rounding to the nearest 100 means looking at the tens digit.1789: tens digit is 8, which is 5 or more, so round up
- Round. The hundreds go up by 1.1789 ≈ 1800
Check. 1,789 is 11 away from 1,800 and 89 away from 1,700. 1,800 is nearer. ✓
Watch out. Write the year in digits first, then round.
Key words. < means “less than” and > means “greater than”. The wide end of the sign faces the bigger number.
Method. Compare the digits from the left, one column at a time. The first column that is different tells you which number is bigger.
- Write each side as one number. We can only compare numbers when they are written the usual way.sixty-two thousand and four = 62,004
62,040 = 62,040 - Write both numbers in a place-value table. Lining up the columns lets us compare digit by digit.
Thousands Ones TTh Th H T O 62,004 6 2 0 0 4 62,040 6 2 0 4 0 - Compare the digits from the left. The first column where they are different decides which number is bigger.ten thousands: 6 and 6 are the same, so look at the next column.
thousands: 2 and 2 are the same, so look at the next column.
hundreds: 0 and 0 are the same, so look at the next column.
tens: 0 and 4 are different. 0 < 4, so 62,004 is smaller. - Write the correct symbol. < means “is less than”, > means “is greater than”, = means “is equal to”.62,004 < 62,040
Check. Picture a number line: 62,004 is to the left of 62,040. ✓
Watch out. Start at the LEFT (the biggest column). Do not start with the ones.
Watch out
Do not stop when you have a number. Check that it answers the question (for example “how many months” or “how far below sea level”), and check it against every clue. Remember to round the real numbers, never numbers that are already rounded.Exercise 1G Problems and puzzles
Non-calculatorMethod. Compare the numbers on the bars. The biggest bar is the park with the most visitors.
- Read each bar. The exact numbers are written on the bars.Park A: 245,000
Park B: 318,500
Park C: 287,000
Park D: 96,200 - Compare from the left. Compare the hundred thousands digits: 2, 3, 2, 0. Park B has the biggest.318,500 > 287,000 > 245,000 > 96,200
- Write the answer. Park B
Check. 318,500 has a 3 in the hundred thousands column. The others have 2 or fewer digits there. ✓
Watch out. Read the numbers on the bars, not just the heights.
Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.
Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.
- Find the two multiples of 10,000 that 96,200 is between. Rounding to the nearest 10,000 means choosing the closest multiple of 10,000.90,000 < 96,200 < 100,000
- Look at the thousands digit (the digit just to the right of the ten thousands digit). This digit tells us if we are past halfway.The thousands digit is 6.
- Decide: round up or round down? 5 or more: round up. 4 or less: round down.6 is 5 or more, so round up.
- Write the rounded number. All the digits to the right of the ten thousands column become zeros.96,200 ≈ 100,000 (to the nearest 10,000)
Check. Distances: 96,200 − 90,000 = 6,200 and 100,000 − 96,200 = 3,800. 100,000 is the nearer one. ✓
Watch out. A 9 can carry: when you round up a 9 it becomes 10, so the next digit goes up too.
Method. Round every number to the nearest 100,000. Then see which ones are the same.
- Round each number. Look at the ten thousands digit each time: 5 or more rounds up, 4 or less rounds down.Park A: 245,000 (digit 4) ≈ 200,000
Park B: 318,500 (digit 1) ≈ 300,000
Park C: 287,000 (digit 8) ≈ 300,000
Park D: 96,200 (digit 9) ≈ 100,000 - Find the numbers that are the same. Park B and Park C both round to 300,000.
- Write the answer. Park B and Park C
Check. The other parks round to different numbers. ✓
Watch out. Round the real numbers, not the rounded ones, each time.
Method. First find the range of numbers that round to 500,000. Then use the cards to make the largest number in that range.
- Find the range for 500,000. To the nearest 100,000, the numbers that round to 500,000 go from 450,000 to 549,999.450,000 to 549,999
- Choose the first digit. The number must be at most 549,999, so the first digit is 5.First digit = 5
- Choose the next digit. The number must still be at most 549,999, so the second digit has to be 4 or less. Take the biggest card that fits.Second digit = 4
- Put the other cards in order, biggest first. This makes the number as big as possible.549,820
Check. 549,820 is between 450,000 and 549,999 and uses every card once. ✓
Watch out. Check your answer rounds to the right number.
Rounded to the nearest 100,000 I am 800,000.
Rounded to the nearest 10,000 I am 840,000.
Rounded to the nearest 1,000 I am 838,000.
My hundreds digit is 4.
My tens digit is double my ones digit, and my ones digit is not 0.
My digits add up to 35.
What number am I?
Method. Use the rounding clues to find the range. Then use the digit clues.
- Use the clue about 1,000. To round to 838,000, the number is from 837,500 to 838,499.837,500 to 838,499
- Check the other rounding clues. 838,000 rounds to 840,000 (10,000) and 800,000 (100,000), so these clues are already true.Range stays 837,500 to 838,499
- Use the hundreds digit. It is 4, so the number is in the range 838,400 to 838,499.838,4_ _
- Use the tens and ones clues. The tens digit is double the ones digit. Try ones digits 1 to 4.ones 1: tens 2: number 838,421, digit total 26
ones 2: tens 4: number 838,442, digit total 29
ones 3: tens 6: number 838,463, digit total 32
ones 4: tens 8: number 838,484, digit total 35 - Use the digit total. The digits add up to 35.Only ones 4 gives a total of 35: 838,484
Check. 838,484: 800,000 ✓, 840,000 ✓, 838,000 ✓, hundreds 4 ✓, tens 8 = 2 × 4 ✓, 8+3+8+4+8+4 = 35 ✓
Watch out. Cross out numbers that do not fit as you use each clue.
Method. Count on in 1,000s, one year at a time. Each year is one jump.
- Say what each jump is. Each year, the amount goes up by 1,000.1 year = +1,000
- Count 5 jumps. Start each jump from the number you have just found.Year 1: 62,500 + 1,000 = 63,500
Year 2: 63,500 + 1,000 = 64,500
Year 3: 64,500 + 1,000 = 65,500
Year 4: 65,500 + 1,000 = 66,500
Year 5: 66,500 + 1,000 = 67,500 - Write the answer. 67,500 seats
Check. 5 × 1,000 = 5,000 and 62,500 + 5,000 = 67,500. ✓
Watch out. Count the jumps carefully. Tally marks can help you keep track.
Method. To find 10,000 less, change the ten thousands digit by 1. Look out for 9s going up (carrying) and 0s going down (swapping).
- Find the ten thousands digit. 10,000 is one ten thousand, so we change the ten thousands digit.
Thousands Ones TTh Th H T O 6 7 5 0 0 - Take 1 from the ten thousands digit. 10,000 less means 1 less ten thousand.6 − 1 = 5 ten thousands
- Write the new number. The digits that did not change stay the same.67,500 − 10,000 = 57,500
Check. Do the opposite: 57,500 + 10,000 = 67,500. ✓
Watch out. Only the digit in the column you are working on changes, unless you carry or swap. Do not change all the digits.
Method. Draw a number line, start at −15 and jump down 15. When you pass 0 the numbers change between positive and negative.
- Show the start on a number line. This lets us see where zero is.
- Go down 15. Split the jump into two parts if it crosses zero: the part that reaches 0, then the part that goes past it.−15 down 15: start at or below zero and go down 15 more → −30
- Write the answer. This is the new depth.−30 m
Check. Do the opposite: −30 + 15 = −15. ✓
Watch out. Going down past zero gives a negative number. Going up from a negative number gets closer to zero.
Method. Find how far apart the two depths are on a number line. Both are below zero.
- Write the depths. The first diver is at −30 m (from part a).−30 m and −11 m
- Find the difference. Both are below zero, so take the smaller distance from the bigger.30 − 11 = 19
- Write the answer. 19 m deeper
Check. −11 − 19 = −30. ✓
Watch out. The difference is how far apart they are. It is never negative.
Key words. An estimate is a sensible answer that is close to the exact one. We find it by rounding first.
Method. Round both numbers to the nearest 1,000, then do the easy calculation.
- Round the first number. Look at the digit just to the right of the thousands column.4,980 ≈ 5,000
- Round the second number. Do the same.5,120 ≈ 5,000
- Do the easy calculation. Rounded numbers are quick to add and take away.5,000 + 5,000 = 10,000
Check. The exact answer is 10,100, which is close to 10,000. ✓
Watch out. An estimate is not exact. It tells you roughly how big the answer should be.
Key words. Roman numerals are letters used for numbers: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000. A small symbol before a bigger one is taken away; after it is added.
Method. Change the Roman numeral to a year, then round it.
- Change MDCCLXXXIX to a number. M = 1,000, D = 500, CC = 200, LXXX = 80, IX = 9.1,000 + 500 + 200 + 80 + 9 = 1789
- Round to the nearest 100. The tens digit is 8, so round up.1789 ≈ 1800
- Write the answer. 1800
Check. 1,800 is only 11 from 1,789. ✓
Watch out. Write the year in digits first, then round.
Method. Find the numbers that could be the real crowd for each newspaper, then see if they overlap.
- Second newspaper. To the nearest 1,000 the crowd is 84,000, so it is from 83,500 to 84,499.83,500 to 84,499
- First newspaper. To the nearest 10,000 the crowd is 80,000, so it is from 75,000 to 84,999.75,000 to 84,999
- Compare. All the numbers from 83,500 to 84,499 are also in the range 75,000 to 84,999.Yes, both can be correct. For example, a crowd of 84,000 rounds to 84,000 and also to 80,000.
Check. 84,000: the thousands digit is 4, so it rounds down to 80,000. ✓
Watch out. Different amounts of rounding can give different numbers for the same crowd.
Method. Write each one as a number and find the one that does not match.
- Write each as a number. Use place value or the Roman numeral rules.1,000 + 200 + 5 = 1,205
1,250
MCCV = 1,000 + 200 + 5 = 1,205
one thousand, two hundred and five = 1,205 - Compare them. Three are 1,205 and one is 1,250.1,205, 1,250, 1,205, 1,205
- Write the odd one out. 1,250
Check. 1,250 has 5 tens. The others have 5 ones. ✓
Watch out. Look at which column the 5 is in.
“When you round a number to the nearest 100, the answer is bigger than the number.”
Method. Try rounding some numbers and compare each answer with the number you started with.
- Try 6,280. The tens digit is 8, so round up.6,280 ≈ 6,300. 6,300 is bigger. It works.
- Try 6,220. The tens digit is 2, so round down.6,220 ≈ 6,200. 6,200 is smaller. It does not work.
- Try 6,300. This is already a multiple of 100.6,300 ≈ 6,300. The same. It does not work.
- Write the answer. It works for some numbers only.Sometimes
Check. Rounding up makes a number bigger, and rounding down makes it smaller. ✓
Watch out. Rounding does not always make a number bigger.
Method. The number must be less than 550,000. Build it from the left, using the biggest digit that fits each time, and never repeat a digit.
- Find the range. To round to 500,000, the number is from 450,000 to 549,999.450,000 to 549,999
- First digit. The biggest first digit that fits is 5.First digit = 5
- Second digit. It must be 4 or less, so that the number is under 550,000. It cannot be 5 because the digits must all be different. The biggest choice is 4.5 4 _ _ _ _
- The other four digits. Use the biggest digits left: 9, 8, 7, 6.5 4 9 8 7 6
- Write the number. 549,876
Check. 549,876 is less than 550,000 and uses six different digits. ✓
Watch out. Check that no digit is used twice.