Ordering Values

Worked example

1. 1

   $4,-7,0,-2,9$ — The list. Some numbers are negative, so picture them on a number line.

2. 2

   $-7,-2$ — Pick out the negatives first. They sit left of zero, so they are the smallest.

3. 3

   $-7<-2$ — −7 is 7 steps below zero and −2 is only 2 below. So −7 is further left, and smaller.

4. 4

   $-7,-2,0$ — Zero comes next. It is bigger than every negative and smaller than every positive.

5. 5

   $-7,-2,0,4,9$ — Last, the positives from small to big: 4, then 9.

Worked example

1. 1

   $0.65,0.6,0.26,0.065,0.206$ — The list. The decimals have different lengths, so they are hard to compare as they are.

2. 2

   $0.650,0.600,0.260$ — The longest have 3 places, so add zeros to the end of the others. Zeros at the end do not change the value.

3. 3

   $0.065,0.206$ — These two already have 3 places, so leave them.

4. 4

   $065<206<260<600<650$ — Now ignore the 0. at the front and compare the digits like whole numbers.

5. 5

   $0.065,0.206,0.26,0.6,0.65$ — Write the answer with the numbers exactly as given, smallest first.

Worked example

1. 1

   $\text{7 digits: B and D}$ — Count the digits. B and D have 7 and the others have 6. More digits means a bigger whole number.

2. 2

   $1141000>1098000$ — B and D both start with 1 million, so look at the next digit: 1 beats 0.

3. 3

   $\text{greatest: D}$ — The question asks which country, so answer with the name D, not the number.

4. 4

   $1141000-912000$ — A 2-mark follow-on asks for the difference between the greatest and the least. A, at 912 000, is the least, so take it from D.

5. 5

   $=229000$ — 1 141 000 − 912 000 = 229 000.

Worked example

1. 1

   $\frac{2}{3},\frac{5}{8},\frac{3}{4},\frac{7}{12}$ — The list. The bottom numbers are all different, so the tops cannot be compared yet.

2. 2

   $\text{common bottom: }24$ — 3, 8, 4 and 12 all go into 24, so every fraction can be written in 24ths.

3. 3

   $\frac{16}{24},\frac{15}{24},\frac{18}{24},\frac{14}{24}$ — Multiply the top and bottom of each by the same number: × 8, × 3, × 6 and × 2. Keep the order of the list.

4. 4

   $14<15<16<18$ — The pieces are now the same size, so the bigger top means the bigger fraction.

5. 5

   $\frac{7}{12},\frac{5}{8},\frac{2}{3},\frac{3}{4}$ — Write the fractions as given, smallest first. 14/24 was 7/12, so it goes first.

Worked example

1. 1

   $\frac{3}{20}\text{and}\frac{4}{20}$ — The question. No whole number lies between the tops 3 and 4, so you cannot answer yet.

2. 2

   $\frac{6}{40}\text{and}\frac{8}{40}$ — Multiply the top and bottom of both by 2. The values stay the same, but now there is a gap between the tops.

3. 3

   $\frac{7}{40}$ — 7 is between 6 and 8, so 7/40 lies between the two fractions. It is exactly halfway too.

Worked example

1. 1

   $0.41,\frac{3}{8},42\%,\frac{2}{5},0.405$ — The list mixes decimals, fractions and a percentage. Change every one to a decimal.

2. 2

   $\frac{3}{8}=3÷8=0.375$ — A fraction means top divided by bottom.

3. 3

   $42\%=42÷100=0.42$ — Percent means out of 100, so divide by 100.

4. 4

   $\frac{2}{5}=2÷5=0.4$ — Top divided by bottom again.

5. 5

   $0.375<0.400<0.405<0.410<0.420$ — Pad them all to 3 places and compare like whole numbers.

6. 6

   $\frac{3}{8},\frac{2}{5},0.405,0.41,42\%$ — Write the original numbers in that order, not the decimals.

Worked example

1. 1

   $35\%\text{ of }360=0.35\times 360$ — 35% is 35 ÷ 100 = 0.35, and “of” means multiply.

2. 2

   $=126$ — 0.35 × 360 = 126. This is the first amount.

3. 3

   $\frac{3}{8}\text{ of }320=320÷8\times 3$ — For 3/8, divide by the bottom to find one eighth, then multiply by the top.

4. 4

   $=40\times 3$ — 320 ÷ 8 = 40. That is one eighth.

5. 5

   $=120$ — 40 × 3 = 120. This is the second amount.

6. 6

   $126>120$ — Compare, then say it in words: 35% of 360 is larger. Show both amounts to get all the marks.

Worked example

1. 1

   $\sqrt{0.3}=\mathrm{0.5477...}$ — Use the calculator for the root. Keep 4 decimal places, not 2.

2. 2

   $\frac{6}{11}=6÷11=\mathrm{0.5454...}$ — A fraction means top divided by bottom.

3. 3

   $55\%=55÷100=0.55$ — Percent means out of 100.

4. 4

   $\frac{\pi }{6}=\mathrm{0.5235...}$ — Type π ÷ 6 on the calculator. Use the π key, not 3.

5. 5

   $5.4\times 10^{-1}=5.4÷10=0.54$ — Times $10^{-1}$ means divide by 10.

6. 6

   $0.5235<0.5400<0.5454<0.5477<0.5500$ — Pad them all to 4 places and compare like whole numbers.

7. 7

   $\frac{\pi }{6},5.4\times 10^{-1},\frac{6}{11},\sqrt{0.3},55\%$ — Write the values back in the form the question used, smallest first.

Worked example

1. 1

   $\frac{3}{8}\square 0.38$ — The question. One is a fraction and one is a decimal, so change the fraction.

2. 2

   $\frac{3}{8}=3÷8=0.375$ — Top divided by bottom.

3. 3

   $0.375\text{and}0.380$ — Pad both to 3 places so the digits line up.

4. 4

   $\frac{3}{8}<0.38$ — 375 is less than 380, so the fraction is smaller. The point of < faces the smaller value.

Worked example

1. 1

   $\frac{9}{40}\square 22.5\%$ — The question. Change both to decimals.

2. 2

   $\frac{9}{40}=9÷40=0.225$ — Top divided by bottom.

3. 3

   $22.5\%=22.5÷100=0.225$ — Percent means divide by 100.

4. 4

   $\frac{9}{40}=22.5\%$ — The decimals are exactly the same, so the answer is =.