The baseball game had a crowd of 3000 fans.
Two thirds of the fans wore a cap.
How many fans did not wear a cap?
Aussie Maths & Science Teachers: Save your time with SmarterEd
The baseball game had a crowd of 3000 fans.
Two thirds of the fans wore a cap.
How many fans did not wear a cap?
`1000`
`text(Fans with no cap)` | `= 1/3 xx 3000` |
`= 1000` |
Nick had a $10 note only.
At the service station, he bought 5 iceblocks that cost $1.20 each.
How much change should he get?
`$4.00` | `$4.50` | `$6.00` | `$8.80` |
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`$4.00`
`text(Change)` | `= 10 – (5 xx 1.20)` |
`= 10 – 6` | |
`= $4.00` |
An unknown number is added to 6.
The result is multiplied by 3 to give an answer of 12.
Which of these is the unknown number?
`− 3` | `− 2` | `2` | `30` |
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`−2`
`text(Working backwards:)`
`text(S)text(ince 4 is multiplied by 3 to make 12,)`
`=>\ text(unknown + 6 = 4)`
`:.\ text(Unknown number = − 2)`
A candy box contains 6 white chocolate bars and 11 dark chocolate bars.
About what percentage of the candy in the box are white chocolate bars?
`6text(%)` | `35text(%)` | `58text(%)` | `65text(%)` |
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`35text(%)`
`text(Percentage of white chocolate bars)`
`=6/17`
`=35text{% (nearest %)}“
Which one of these numbers is a factor of 34?
`3` | `4` | `17` | `68` |
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`17`
`17`
Bonny draws the number 1 or 2 on a group of disks, pictured below.
One disk is chosen at random.
What is the chance the disk has a 2 drawn on it?
`1/3` | `5/15` | `5/18` | `13/18` |
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`5/18`
`text(Fraction with a “2”)` | `= text(Number of 2’s) / text(Total circles)` |
`=5/18` |
Nigella had 1 kilogram of chocolate.
She used `3/4` kilogram in a cake recipe.
How much chocolate did Nigella have left?
`0.75\ text(kg)` | `0.70\ text(kg)` | `0.33\ text(kg)` | `0.25\ text(kg)` |
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`0.25\ text(kg)`
`text(Chocolate left)` | `=1 – 3/4` |
`= 1/4` | |
`= 0.25\ text(kg)` |
Peter had 5 cups of flour to use for baking 2 loaves of bread.
He used `1 1/8` cups for the first loaf, and `2 1/8` cups for the second loaf.
How many cups of flour did Peter have left?
`1 3/4` | `1 1/2` | `2 1/2` | `8 1/4` |
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`1 3/4`
`text(Cups of flour left)`
`=5 – (1 1/8 + 2 1/8)`
`= 5- 3 1/4`
`=1 3/4`
A conservationist is studying gorillas in Uganda.
Which unit would be the most appropriate to record the mass of a gorilla?
`text(gram)` | `text(kilogram)` | `text(litre)` | `text(millimetre)` |
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`text(kilogram)`
`text(A gorilla will weigh hundreds of kilograms.)`
`:.\ text(The best unit is kilogram.)`
Penelope is making a pattern with black and white tiles, as shown below.
If the pattern continues, how many black tiles will there be in Design 8?
`4` | `16` | `32` | `44` |
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`32`
`text(Each design adds 4 black tiles.)`
`:.\ text(Black tiles in design 8)` | `=8 xx 4` |
`=32` |
Kate just turned 11 years old.
Patrick is 3 years older than twice Kate's age.
How old is Patrick?
`28` | `56` | `31` | `25` |
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`25`
`text(Patrick’s age)` | `= (11 xx 2) + 3` |
`= 25` |
Tim is selling milkshakes at a school fete.
He makes $48 from selling 8 milkshakes.
All milkshakes cost the same.
How much will Tim make if he sells 11 milkshakes.
`$43` | `$63` | `$66` | `$88` |
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`$66`
`text(Price of 1 milkshake)` | `=($48)/8=$6` |
`:.\ text(Price of 11 milkshakes)` | `=11 xx $6` |
`=$66` |
Oliver's karate grading started at 9:15 am and went for `3 1/4` hours.
He then went straight home, which took him `3/4` hours.
What time did Oliver get home?
`text(12:30 pm)` | `text(1:00 am)` | `text(1:00 pm)` | `text(1:15 am)` | `text(1:15 pm)` |
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`text(1:15 pm)`
`text(Karate finishes at 9:15 plus 3 hours and 15)`
`text(minutes, which is 12:30 pm.)`
`text(12:30 plus 45 minutes means Oliver arrives)`
`text(home at 1:15 pm.)`
The first number in a pattern is 3.8.
Each number in the pattern is formed by subtracting 0.35 from the previous number.
What is the third number in this pattern?
`2.75` | `3.1` | `3.45` | `4.5` |
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`3.1`
`3.8 – 0.35 = 3.45`
`3.45 – 0.35 = 3.1`
The number of students that could solve a Rubik's cube in less than 2 minutes were counted at 7 different primary schools.
The results were recorded in the graph below.
How many students, in total, could solve the cube in less than 2 minutes?
`3` | `14` | `15` | `18` |
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`15`
`text(Total students who can solve in less than 2 mins)`
`= 2+2+3+0+3+2+3`
`=15`
Peter has a marble bag that contains 20 marbles that are either red or green in colour.
The probability of randomly picking a green marble is 70%.
What is the probability of randomly picking a red marble?
% |
`text(30%)`
`P(text(green)) + P(text(red))` | `= 100text(%)` |
`text(70%)\ + P(text(red))` | `= 100text(%)` |
`:. P(text(red))` | `= 30text(%)` |
Michelle purchased the 4 items pictured below at the supermarket.
The total mass of Michelle's items is closest to
`4\ text(kg)` | `5\ text(kg)` | `8\ text(kg)` | `9\ text(kg)` |
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`text(4 kg)`
`text(Total mass)` | `=0.2+1+0.250+3` |
`=4.450\ text(kg)` |
`:.\ text(Closest mass is 4 kg.)`
An orientation course is set up with 4 drinking stations.
Runners start and then progress through the course in the order of the drinking stations labelled 1-4 in the diagram below.
At which drinks station do the runners make the greatest change of direction?
Station 1 | Station 2 | Station 3 | Station 4 |
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`text(Station 2)`
`text(Station 2)`
John wants to find the answer to `2998 + 1567`.
Which of the following shows a way to get the same answer?
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`3000 + 1566` |
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`3000 + 1565` |
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`3004 + 1500` |
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`3004 + 1600` |
`3000 + 1565`
`text(By adding and subtracting 2:)`
`2998+1567` | `=2998+2+1567-2` |
`=3000 + 1565` |
Rahul draws a net, which is pictured below.
What 3D object will it make?
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Pentagonal prism |
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Pentagonal pyramid |
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Hexagonal pyramid |
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Hexagonal prism |
`text(Pentagonal pyramid)`
`text(The net will have a pentagon shaped base, and)`
`text(converge to a point.)`
`:.\ text(Net will make a pentagonal pyramid.)`
At an apple orchard, apples are picked and put in a basket.
The table below shows the total number of apples in the basket after each minute.
\begin{array} {|c|c|c|}
\hline \textbf{Minutes} & \textbf{Total number of apples} \\
\hline 1 & 4 \\
\hline 2 & 8 \\
\hline 3 & 12 \\
\hline 4 & 16 \\
\hline \end{array}
How many apples are in the basket after 10 minutes?
`20` | `30` | `35` | `40` |
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`40\ text(apples)`
`text(4 apples are put into the basket each minute.)`
`:.\ text(Apples in basket after 10 minutes)`
`=4 xx 10`
`= 40\ text(apples)`
Zilda has 4 children.
Each child needs 6 pens when they start school for the new year.
Zilda has no pens when she goes to the shop.
The shop sells its pens in packets of 5.
How many packets does Zilda need to buy?
`2` | `4` | `5` | `6` |
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`5\ text(packets)`
`text(Pens required)` | `= 4 xx 6` |
`= 24` |
`:.\ text(Packets required)` | `= 24/5` |
`= 4.8` | |
`= 5` |
Anne taught a kindergarten class and asked her students what their favorite fruit is.
Their answers were used to draw the pie chart below.
Around a quarter of her students preferred which fruit?
`text(Banana)` | `text(Apple)` | `text(Mango)` | `text(Grapes)` |
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`text(Apple)`
`text(A quarter of students will represent a sector that)`
`text(is the size of a quarter of the circle.)`
`:.\ text(Apple)`
The airforce buys 8 new fighter jets for $882.0 million.
Each jet costs the same amount.
What is the cost of 1 fighter jet?
`$110.25\ text(million)` | `$441.0\ text(million)` | `$874.0\ text(million)` | `$7056\ text(million)` |
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`$110.25\ text(million)`
`{$882.0\ text(million)}/ 8 = $110.25\ text(million)`
Which expression is equal to `4a^4`?
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`8 xx a` |
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`16 xx a` |
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`4 xx a xx a xx a xx a` |
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`4 + a + a + a + a` |
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`4 xx 4 xx 4 xx 4 xx a xx a xx a xx a` |
`4 xx a xx a xx a xx a`
`4a^4 = 4 xx a xx a xx a xx a`
Bogdan grows large turnips in his garden.
One turnip he picks weighs 1.2 kg.
Bogdan cuts 300 grams off the turnip to cook with.
What fraction has he used for cooking?
`1/2` | `1/3` | `1/4` | `1/5` |
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`1/4`
`text(Fraction)` | `=300/1200` |
`=1/4` |
Cameron paid $1.44 for 9 pencils.
How much did each pencil cost?
`$0.12` | `$0.16` | `$1.35` | `$1.53` |
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`$0.16`
`text(C)text(ost of 1 pencil)` | `= 1.44/9` |
`=$0.16` |
David buys a surfboard cover online that costs him $128.
The postage costs involved are listed in the table below.
The surfboard cover weighs 1.1 kg
What does David have to pay, in total, to purchase the surfboard and have it delivered?
`$140` | `$145.50` | `$149.75` | `$153` |
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`$149.75`
`text(Surfboard cover cost)\ +\ text(postage)`
`= $128 + $21.75`
`= $149.75`
Nigella placed a leg of lamb into her oven in the morning.
The 1st measurement of the lamb's temperature was −6°C.
Fifteen minutes later, a second temperature reading was taken, which measured 16°C.
What was the change in temperature between the two measurements?
`text(decrease of)\ 22^@text(C)` | `text(decrease of)\ 10^@text(C)` | `text(increase of)\ 10^@text(C)` | `text(increase of)\ 22^@text(C)` |
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`text(increase of)\ 22^@text(C)`
`text(Change in temperature)` | `=16-(-6)` |
`=22^@ text(C)` |
`:.\ text(An increase of 22°C.)`
A circle is divided into 5 equal areas and labelled, as shown in the diagram below.
What percentage of the circle's area has been labelled with an odd number?
`text(3%)` | `text(15%)` | `text(30%)` | `text(40%)` | `text(60%)` |
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`text(60%)`
`text(S)text(ince all areas are equal,)`
`text(Percentage labelled with an odd number)`
`=3/5`
`=60text(%)`
9 cubes are used to make an obstacle, pictured below, that is part of a children's adventure playground.
Gary paints all sides of the step except for the bottom side which sits on the ground.
How many cube faces will Gary paint?
`15` | `18` | `24` | `30` | `36` |
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`24`
`24`
The first three days of the Brisbane cricket test had the following attendances:
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex}\text{Day 1}\rule[-1ex]{0pt}{0pt} & 20\ 156\\
\hline
\rule{0pt}{2.5ex}\text{Day 2}\rule[-1ex]{0pt}{0pt} & 18\ 397\\
\hline
\rule{0pt}{2.5ex}\text{Day 3}\rule[-1ex]{0pt}{0pt} & 29\ 981\\
\hline
\end{array}
What was the total crowd over the first 3 days, to the nearest `1000?`
`68\ 000` | `69\ 000` | `70\ 000` | `71\ 000` |
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`69\ 000\ text{(nearest 1000)}`
`20\ 156 + 18\ 397 + 29\ 981`
`= 68\ 534`
`= 69\ 000\ text{(nearest 1000)}`
`2.5 xx A = 0.5`
Find the value of `A` that makes this number sentence correct.
`1.25` | `-2.0` | `0.2` | `5.0` |
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`0.2`
`2.5 xx A` | `= 0.5` |
`:. A` | `= 0.5/2.5` |
`= 0.2` |
A giant earthworm measures 2.1 metres.
How long is it in centimetres?
`201` | `210` | `2010` | `2100` |
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`text(210 cm)`
`text{Length (cm)}` | `= 2.1 xx 100` |
`= 210\ text(cm)` |
Ben spent twice as much money as Mark.
If they spent a total of $90, how much did Ben spend?
`$15` | `$30` | `$60` | `$180` |
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`$60`
`text(Strategy 1)`
`text(Ratio)`
`text{Ben : Mark = 2 : 1 (3 parts)}`
`text(1 part)\ = $90-:3=$30`
`:.\ text(Ben spent 2 × $30 = $60)`
`text(Strategy 2)`
`text(Let)\ \ $x = text(Amount Mark spent)`
`2x + x` | `= 90` |
`x` | `= $30` |
`:.\ text(Ben spent $60.)`
Andreas has $8 to buy batteries for his toy racing car.
Each battery costs $1.60 and he buys 4 batteries.
Which expression shows how much money he has left?
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`$8 - $1.60` |
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`$8 - (4 xx $1.60)` |
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`$8 - $1.60 + $1.60 + $1.60 + $1.60` |
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`4 xx ($8 - $1.60)` |
`$8 – (4 xx $1.60)`
`$8 – (4 xx $1.60)`
Janus measures the width of his driveway to be 4 metres and 18 centimetres.
Which answer shows how Janus can write this measurement in metres?
`4.018\ text(m)` | `4.18\ text(m)` | `4.1\ text(m)` | `5.18\ text(m)` |
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`text(4.18 m)`
`text(Measurement = 4.18 m)`
Peter left home at 9:15 in the morning and did not return until 5:25 in the afternoon.
How long was Peter away from his house?
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3 hours 50 minutes |
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4 hours 10 minutes |
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7 hours 50 minutes |
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8 hours 10 minutes |
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13 hours 20 minutes |
`8\ text(hours)\ 10\ text(minutes)`
`text(Time away)` | `= 2 text(h)\ 45\ text(min) + 5 text(h)\ 25\ text(min)` |
`= 8\ text(hours)\ 10\ text(minutes)` |
On a country property, 1 acre of land is recommended for every 4 sheep.
How many acres of land would be needed for 16 sheep?
`4` | `6` | `16` | `64` |
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`4\ text(acres)`
`text(Acres)` | `= 16/4` |
`=4` |
Which of the following is equal to 32?
`2^3 xx 2^2` | `2^3 + 2^2` | `3^2 + 2^2` | `3^2 xx 2^2` |
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`2^3 xx 2^2`
`2^3 xx 2^2` | `= 8 xx 4` |
`= 32` |
The point `T(2at,at^2)` lies on the parabola `P_1` with the equation `x^2=4ay`.
The tangent to the parabola `P_1` at `T` meets the directrix at `D`.
The normal to the parabola `P_1` at `T` meets the vertical line through `D` at the point `R`, as shown in the diagram.
It can be shown that the minimum distance between `R` and `T` occurs when the normal to `P_1` at `T` is also the normal to `P_2` at `R`. (Do NOT prove this.)
i. `text(Show)\ \ D (text{at}\ -a/t, -a)`
`text(T)text(angent equation at)\ \ T:`
`y = tx – at^2`
`D\ \ text(occurs when)\ \ y = -a,`
`tx – at^2` | `= -a` |
`tx` | `= at^2 – a` |
`x` | `= at – a/t` |
`:. D\ text(has coordinates)\ \ (text{at}\ -a/t, -a)`
ii. `text(Normal equation at)\ \ T:`
`x + ty = 2at + at^3`
`R\ text(occurs when)\ \ x = at – a/t`
`at – a/t + ty` | `= 2at + at^3` |
`ty` | `= at + a/t + at^3` |
`y` | `= a + a/t^2 + at^2` |
`= a(1 + 1/t^2 + t^2)\ \ …\ text{(*)}` |
`:. R (a (t – 1/t), a (1 + 1/t^2 + t^2))`
`x^2` | `= a^2 (t – 1/t)^2` |
`= a^2 (t^2 – 2 + 1/t^2)` | |
`= a + (1+1/t^2 + t^2 – 3)` | |
`= a^2 (y/a – 3)\ \ text{(see (*) above)}` | |
`= ay – 3a^2` |
`:.\ text(Locus of)\ R\ text(is)\ \ x^2 = ay – 3a^2`
iii. `text(In the form)\ \ x^2 = 4ay,`
`x^2` | `= a(y – 3a)` |
`= 4 · a/4 (y – 3a)` |
`:.\ text(Focal length) = a/4`
iv. `text(Equation of)\ \ P_2`
`x^2` | `= ay – 3a^2` |
`y` | `= x^2/a + 3a` |
`y prime` | `= (2x)/a` |
`text(At)\ \ R,\ \ x = a (t – 1/t)`
`:.\ text(Gradient of normal at)\ \ R`
`=(-a)/(2x)`
`= (-a)/(2a(t – 1/t)) xx t/t`
`= (-t)/(2(t^2 – 1))`
`text(Gradient of normal at)\ \ T:`
`x + ty` | `= 2at + at^3` |
`y` | `= -1/t x + 2a + at^2` |
`:. m` | `= -1/t` |
`text(Distance)\ \ RT\ \ text(is a minimum when)`
`(-t)/(2(t^2 – 1))` | `= -1/t` |
`t^2` | `= 2t^2 – 2` |
`t^2` | `= 2` |
`:. t` | `= +- sqrt 2` |
Consider the expansion of `(1 + x)^n`, where `n` is a positive integer.
i. `text(Show)`
`2^n = ((n),(0)) + ((n),(1)) + ((n),(2)) + ((n),(3)) + … + ((n),(n))`
`text(Using binomial expansion)`
`(1 + x)^n = ((n), (0)) + ((n), (1))x + ((n),(2)) x^2 + … + ((n), (n)) x^n`
`text(Let)\ \ x = 1,`
`2^n = ((n), (0)) + ((n), (1)) + ((n), (2)) + … + ((n), (n))`
`text(… as required.)`
ii. `text(Differentiate both sides of expansion,)`
`n (1 + x)^(n – 1) = ((n), (1)) + 2 ((n), (2))x + 3 ((n), (3)) x^2 + … + n ((n), (n)) x^(n – 1)`
`text(Let)\ \ x = 1,`
`n2^(n – 1) = ((n), (1)) + 2 ((n), (2)) + 3 ((n), (3)) + … + n ((n), (n))`
`text(… as required.)`
iii. `text{Multiply part (i)} xx n`
`n2^n` | `= n[((n), (0)) + ((n), (1)) + … + ((n), (n))]` |
`= sum_(r = 0)^n ((n), (r)) n\ \ text{… (1)}` |
`text{Multiply part (ii)} xx 2`
`2 xx n2^(n – 1)` | `= 2[((n), (1)) + 2 ((n), (2)) + … + n ((n), (n))]` |
`n2^n` | `= sum_(r = 1)^n ((n), (r)) 2r\ \ text{… (2)}` |
`text(Subtract) qquad (2) – (1)`
`sum_(r = 1)^n ((n), (r)) 2r – sum_(r = 0)^n ((n), (r))n` | `= n2^n – n2^n` |
`sum_(r = 1)^n ((n), (r)) 2r – sum_(r = 1)^n ((n), (r)) n – n` | `= 0` |
`sum_(r = 1)^n ((n), (r)) (2r – n)` | `= n\ \ text(… as required)` |