Compare quantities accurately by first writing them in a common unit or common number form. This chapter develops ordering from integers and decimals to fractions, percentages and mixed-form exam questions.
Subtopic 1
On a number line, numbers further to the right are greater. Negative numbers can cause mistakes: for example, \(-2>-7\) because \(-2\) lies further right.
If quantities use different units, first convert them all to the same unit. It is often easiest to use the smallest unit so you can compare whole-number values.
Always use one unit before comparing quantities; retain signs when heights are measured above and below a reference level.
Write \(-3^\circ\text{C},\ 5^\circ\text{C},\ -8^\circ\text{C},\ 0^\circ\text{C}\) in ascending order.
From smallest (furthest left on the number line) to largest:
\[\boxed{-8^\circ\text{C},\ -3^\circ\text{C},\ 0^\circ\text{C},\ 5^\circ\text{C}}\]Put these in order, smallest first: \(1.8\text{ m},\ 145\text{ cm},\ 920\text{ mm},\ 2.05\text{ m}\).
Convert all to millimetres:
\[1800,\ 1450,\ 920,\ 2050\text{ mm}\]So the original quantities are:
\[\boxed{920\text{ mm},\ 145\text{ cm},\ 1.8\text{ m},\ 2.05\text{ m}}\]Order \(0.75\text{ L},\ 68\text{ cl},\ 725\text{ ml},\ 0.8\text{ L}\) from greatest to smallest.
Convert to millilitres:
\[750,\ 680,\ 725,\ 800\text{ ml}\]Greatest to smallest:
\[\boxed{0.8\text{ L},\ 0.75\text{ L},\ 725\text{ ml},\ 68\text{ cl}}\]A balloon rises from \(-125\text{ m}\) to \(85\text{ m}\). How much higher is its final position?
Start at \(-125\) and move up to zero: \(125\text{ m}\).
Then move another \(85\text{ m}\) above zero.
\[125+85=\boxed{210\text{ m}}\]
Try each question before opening its worked solution.
Arrange \(-11,-4,0,-9,6\) from smallest to greatest.
Negative values lie below zero; \(-11\) is furthest left.
\[\boxed{-11,-9,-4,0,6}\]
Put \(0.75\text{ kg},\ 680\text{ g},\ 1.2\text{ kg}\) in ascending order.
Convert to grams: \(750,680,1200\).
\[\boxed{680\text{ g},0.75\text{ kg},1.2\text{ kg}}\]
Three underwater markers are \(2.4\text{ m}\) below sea level, \(185\text{ cm}\) below sea level and \(0.7\text{ m}\) above it. Order their signed heights from lowest to highest.
Using metres: \(-2.4,-1.85,+0.7\).
\[\boxed{-2.4\text{ m},-1.85\text{ m},+0.7\text{ m}}\]
Subtopic 2
Inequality symbols compare values. Read the symbol from left to right.
| Symbol | Meaning | Example |
|---|---|---|
| \(<\) | less than | \(-4<3\) |
| \(>\) | greater than | \(7>2\) |
| \(\leq\) | less than or equal to | \(x\leq5\) |
| \(\geq\) | greater than or equal to | \(y\geq-1\) |
| \(=\) | equal to | \(0.5=\frac12\) |
| \(\neq\) | not equal to | \(0.3\neq3\%\) |
For \(\le\) or \(\ge\), equality is allowed; for \(<\) or \(>\), the boundary is excluded.
Insert \(<\) or \(>\): \(-6\ \square\ -2\).
\(-6\) is further left on the number line, so it is smaller.
\[\boxed{-6<-2}\]A score must be at least \(18\) but below \(25\). Write a combined inequality for score \(s\).
“At least 18” means \(s\ge18\), so 18 is included.
“Below 25” means \(s<25\), so 25 is excluded.
\[\boxed{18\le s<25}\]
Among negative numbers, the value with the larger-looking magnitude is actually smaller: \(-12<-5\).
Try each question before opening its worked solution.
Insert \(<\), \(>\) or \(=\): \(-0.4\ \square\ -0.37\).
Write \(-0.4=-0.40\). Since \(-0.40\) is further left,
\[\boxed{-0.4<-0.37}\]
Write “the temperature is no higher than \(3^\circ\text{C}\)” as an inequality for \(T\).
No higher means 3 or any lower value.
\[\boxed{T\le3}\]
A delivery accepts a mass greater than \(2.5\text{ kg}\) but no more than \(5\text{ kg}\). Represent the permissible mass \(m\).
2.5 is excluded; 5 is included.
\[\boxed{2.5<m\le5}\]
Subtopic 3
Compare decimals using place value. Adding zeros to the end does not change a decimal's value, so it can make comparisons much clearer.
For example, write \(0.7\) as \(0.700\) if you are comparing it with \(0.706\) and \(0.069\). Then compare digits from left to right.
| Number | Tenths | Hundredths | Thousandths |
|---|---|---|---|
| 0.706 | 7 | 0 | 6 |
| 0.700 | 7 | 0 | 0 |
| 0.069 | 0 | 6 | 9 |
Adding zeros at the end of a decimal does not alter its value. In negative comparisons, the number closer to zero is greater.
Put \(0.0408,\ 0.408,\ 0.0048,\ 0.048\) in ascending order.
Write each to four decimal places:
\[0.0408,\ 0.4080,\ 0.0048,\ 0.0480\]Therefore:
\[\boxed{0.0048<0.0408<0.048<0.408}\]Arrange \(6.02,\ 6.2,\ 6.002,\ 6.022,\ 6.2001\) from greatest to smallest.
Write to four decimal places:
\[6.0200,\ 6.2000,\ 6.0020,\ 6.0220,\ 6.2001\]So:
\[\boxed{6.2001>6.2>6.022>6.02>6.002}\]Order \(5.07,5.007,5.7,5.0705\) from least to greatest.
Add trailing zeros to compare positions: \(5.0700,5.0070,5.7000,5.0705\).
Compare tenths, then hundredths, thousandths and so on.
\[\boxed{5.007<5.07<5.0705<5.7}\]
Try each question before opening its worked solution.
Order \(0.5,0.055,0.505,0.0505\) from greatest to least.
Compare as \(0.5000,0.0550,0.5050,0.0505\).
\[\boxed{0.505>0.5>0.055>0.0505}\]
Which is larger: \(-2.03\) or \(-2.3\)?
Use \(-2.03\) and \(-2.30\); the number nearer zero is greater.
\[\boxed{-2.03>-2.3}\]
Four speeds are \(7.09,7.9,7.099,7.009\text{ m/s}\). List them in ascending order.
Use three decimal places: \(7.090,7.900,7.099,7.009\).
\[\boxed{7.009,7.09,7.099,7.9\text{ m/s}}\]
Subtopic 4
There are two reliable methods. Without a calculator, use a common denominator. With a calculator, converting fractions to decimals can be efficient.
For exact comparison use a common denominator; avoid early rounding of recurring-decimal equivalents.
Which is larger, \(\frac58\) or \(\frac23\)?
LCM of 8 and 3 is 24.
\[\frac58=\frac{15}{24},\qquad\frac23=\frac{16}{24}\]So:
\[\boxed{\frac23>\frac58}\]Put \(\frac7{10},\frac58,\frac34,\frac9{20}\) in ascending order.
Use denominator 40:
\[\frac7{10}=\frac{28}{40},\quad\frac58=\frac{25}{40},\quad\frac34=\frac{30}{40},\quad\frac9{20}=\frac{18}{40}\]Therefore:
\[\boxed{\frac9{20}<\frac58<\frac7{10}<\frac34}\]Put \(\frac{11}{16},\frac{17}{24},\frac{7}{12}\) in descending order.
A common denominator is 48:
\[\frac{11}{16}=\frac{33}{48},\quad\frac{17}{24}=\frac{34}{48},\quad\frac7{12}=\frac{28}{48}\]So:
\[\boxed{\frac{17}{24}>\frac{11}{16}>\frac7{12}}\]Arrange \(\frac7{12},\frac35,\frac58\) in descending order.
The LCM of 12, 5 and 8 is 120.
\[\frac7{12}=\frac{70}{120},\quad\frac35=\frac{72}{120},\quad\frac58=\frac{75}{120}\]
\[\boxed{\frac58>\frac35>\frac7{12}}\]
Try each question before opening its worked solution.
Which is larger, \(\frac5{12}\) or \(\frac38\)?
Common denominator 24: \(\frac5{12}=\frac{10}{24}\), \(\frac38=\frac9{24}\).
\[\boxed{\frac5{12}>\frac38}\]
Arrange \(\frac49,\frac12,\frac5{12}\) in ascending order.
Using denominator 36 gives \(\frac{16}{36},\frac{18}{36},\frac{15}{36}\).
\[\boxed{\frac5{12}<\frac49<\frac12}\]
Three students completed \(\frac{11}{20},\frac59,\frac7{12}\) of a task. Rank the fractions from largest to smallest.
Using denominator 180: \(\frac{99}{180},\frac{100}{180},\frac{105}{180}\).
\[\boxed{\frac7{12}>\frac59>\frac{11}{20}}\]
Subtopic 5
When forms are mixed, convert them all to one form before comparing. Decimal form is usually the quickest.
Divide by 100. For example, \(7\%=0.07\), not 0.7.
Calculate numerator \(\div\) denominator.
First change every value to one form, compare accurately, then write the order using the original representations.
Put \(\frac38,\ 41\%,\ 0.39,\ \frac25\) in ascending order.
Convert to decimals:
\[\frac38=0.375,\quad41\%=0.41,\quad0.39=0.39,\quad\frac25=0.4\]Therefore:
\[\boxed{\frac38<0.39<\frac25<41\%}\]Three teams have all played the same number of matches. Team A won \(\frac{11}{20}\), Team B won 54%, and Team C won 0.57 of their matches. Rank them from most successful to least successful.
Convert to decimals:
\[\frac{11}{20}=0.55,\quad54\%=0.54,\quad0.57=0.57\]So:
\[\boxed{\text{Team C, Team A, Team B}}\]Arrange \(62\%,\ 0.605,\ \frac35,\ 0.62\) in ascending order. Identify equal values.
Convert to decimals: \(62\%=0.62\) and \(\frac35=0.6\).
Compare \(0.6,0.605,0.62,0.62\).
\[\boxed{\frac35<0.605<62\%=0.62}\]
After converting to compare, write your final answer in the original forms unless the question asks otherwise.
Try each question before opening its worked solution.
Order \(\frac14,23\%,0.3\) from smallest to largest.
Convert: \(0.25,0.23,0.30\).
\[\boxed{23\%<\frac14<0.3}\]
Which is larger, \(\frac7{20}\) or \(34.8\%\)?
\(\frac7{20}=0.35=35\%\), while \(34.8\%=0.348\).
\[\boxed{\frac7{20}>34.8\%}\]
Four candidates receive \(\frac38,37\%,0.375,\frac7{20}\) of a vote. Put the shares in descending order and identify a tie.
Decimal equivalents: \(0.375,0.37,0.375,0.35\).
\[\boxed{\frac38=0.375>37\%>\frac7{20}}\]
The early questions build confidence with negatives and decimals. Later questions mix units and number forms in the way examination questions often do.
Write \(-4,\ 7,\ -9,\ 0,\ 3\) in ascending order.
From smallest to largest:
\[\boxed{-9,-4,0,3,7}\]Insert the correct symbol \(<, >, =\) or \(\neq\).
a) \(-5>-11\)
b) \(0.25=\frac14\)
c) \(8\%=0.08\), so \(8\%\neq0.8\)
d) \(-2<0\)
Arrange \(0.405,\ 0.045,\ 0.450,\ 0.054\) from smallest to largest.
Write all to three decimal places:
\[0.405,0.045,0.450,0.054\]So:
\[\boxed{0.045<0.054<0.405<0.450}\]Which is larger, \(\frac7{12}\) or \(\frac35\)?
Use denominator 60:
\[\frac7{12}=\frac{35}{60},\qquad\frac35=\frac{36}{60}\]Therefore \(\boxed{\frac35}\) is larger.
Put \(2.6\text{ m},\ 245\text{ cm},\ 2700\text{ mm},\ 2.55\text{ m}\) in ascending order.
Convert to mm:
\[2600,2450,2700,2550\]So:
\[\boxed{245\text{ cm},\ 2.55\text{ m},\ 2.6\text{ m},\ 2700\text{ mm}}\]Order \(3.2\text{ kg},\ 3180\text{ g},\ 0.31\text{ kg},\ 3050\text{ g}\) from greatest to smallest.
Convert to grams:
\[3200,3180,310,3050\]Greatest to smallest:
\[\boxed{3.2\text{ kg},\ 3180\text{ g},\ 3050\text{ g},\ 0.31\text{ kg}}\]Arrange \(\frac23,\frac7{10},\frac58,\frac34\) from smallest to largest.
Use denominator 120:
\[\frac23=\frac{80}{120},\ \frac7{10}=\frac{84}{120},\ \frac58=\frac{75}{120},\ \frac34=\frac{90}{120}\]So:
\[\boxed{\frac58<\frac23<\frac7{10}<\frac34}\]Put \(4.071,\ 4.7,\ 4.017,\ 4.107,\ 4.0701\) in ascending order.
Write to four decimal places:
\[4.0710,4.7000,4.0170,4.1070,4.0701\]So:
\[\boxed{4.017<4.0701<4.071<4.107<4.7}\]Arrange \(\frac9{20},\ 46\%,\ 0.44,\ \frac{11}{25}\) from smallest to largest.
Convert to decimals:
\[\frac9{20}=0.45,\quad46\%=0.46,\quad0.44=0.44,\quad\frac{11}{25}=0.44\]So:
\[\boxed{0.44=\frac{11}{25}<\frac9{20}<46\%}\]Put \(62\%,\frac58,\ 0.61,\frac35\) in descending order.
Convert to decimals:
\[0.62,\quad0.625,\quad0.61,\quad0.6\]So:
\[\boxed{\frac58>62\%>0.61>\frac35}\]Order \(0.333,\ \frac13,\ 33.4\%\) from smallest to largest.
Convert to decimals:
\[0.333,\quad\frac13=0.3333\ldots,\quad33.4\%=0.334\]Therefore:
\[\boxed{0.333<\frac13<33.4\%}\]At four locations the temperatures are \(-1.8^\circ\text{C},\ 0.4^\circ\text{C},\ -2.05^\circ\text{C},\ -1.75^\circ\text{C}\). List the temperatures from warmest to coldest.
Warmest means greatest. Compare:
\[0.4>-1.75>-1.8>-2.05\]So:
\[\boxed{0.4^\circ\text{C},\ -1.75^\circ\text{C},\ -1.8^\circ\text{C},\ -2.05^\circ\text{C}}\]In a survey, \(\frac{5}{16}\) chose A, 30% chose B, and 0.32 chose C. Rank A, B and C from most popular to least popular.
Convert to decimals:
\[\frac5{16}=0.3125,\quad30\%=0.30,\quad0.32=0.32\]So:
\[\boxed{\text{C, A, B}}\]Four ropes have lengths \(1.85\text{ m},\ 184\text{ cm},\ 1865\text{ mm},\ 1.805\text{ m}\). Write them in order from shortest to longest.
Convert to mm:
\[1850,1840,1865,1805\]So:
\[\boxed{1.805\text{ m},\ 184\text{ cm},\ 1.85\text{ m},\ 1865\text{ mm}}\]The numbers are \(47\%,\frac{12}{25},\ 0.465,\frac9{20},\ 0.49\). Put them in ascending order and state the median value.
Convert to decimals:
\[47\%=0.47,\quad\frac{12}{25}=0.48,\quad0.465,\quad\frac9{20}=0.45,\quad0.49\]Ascending order:
\[\boxed{\frac9{20}<0.465<47\%<\frac{12}{25}<0.49}\]The middle value is \(\boxed{47\%}\).
I use a number-line idea and know that the more negative value is smaller.
I convert all measurements to one unit before comparing.
I use place value and can add trailing zeros without changing a value.
I can compare using a common denominator or decimal conversion.
I convert fractions, decimals and percentages to one common form before ordering.
I restore the original forms and units after comparing.
Illustrations are taken only from actual Chapter 5 diagrams where present; explanations and practice questions are newly written. Text-only book pages are not reproduced as images.