Reigate Grammar School Paper 1

IndependentCo-ed

23 questions

Original PDF

Questions

1. 1
  If an adult ticket on the bus costs $\pounds 1.50$ and a child ticket costs $90 \ \text{p}$ , how much change will a family of $2$ adults and their $3$ children get if they pay with a $\pounds 10$ note?
  Keep in mind
  It is really important to use the right units in calculations and convert where required
  $\begin{aligned} \text {Money spent} &= 2 \times \pounds 1.50 + 3 \times 90 \ \text{p} \\\\ &= \pounds 3 + 3 \times \pounds 0.90 \\\\ &= 3 + 2.70 \\\\ &= \pounds 5.70 \\\\ \text {Change received} &= 10 - 5.70 \\\\ &= \pounds 4.30 \end{aligned}$
  Answer
  $\pounds 4.30$
2. 2
  If a square has an area of $36 \ \text{cm}^2$ , what is its perimeter?
  Formulas used
  $\text{Area of square} = \text{side} \times \text{side}$
  $\text{Perimeter of square} = 4 \times \text{side}$
  Define variables
  Let the side of the square (in cm) $= a$
  
  Side length
  From the question,
  $\begin{aligned} a \times a &= 36 \\\\ a^{2} &= 36 \\\\ a &= \sqrt {36} \\\\ a &= 6 \ \text{cm} \end{aligned}$
  
  Answer
  $\begin{aligned} \text{Side} &= 6 \ \text{cm} \\\\ \text{Perimeter} &= 4 \times 6 \\\\ &= 24 \ \text{cm} \end{aligned}$
  Answer
  $24 \ \text{cm}$
3. 3
  What is $0.007$ written as a fraction?
  $\begin{aligned} &= 0.007 \\\\ &= \dfrac {7} {1000} \end{aligned}$
  Answer
  $\dfrac {7} {1000}$
4. 4
  How many $\text{minutes}$ are there in one day?
  Formulas used
  $1 \ \text{hour} = 60 \ \text{minutes}$
  $\begin{aligned} &= 1 \ \text{day} \\\\ &= 1 \times 24 \ \text{hours} \\\\ &= 24 \times 60 \ \text{minutes} \\\\ &= 1440 \ \text{minutes} \end{aligned}$
  Answer
  $1440 \ \text{minutes}$
5. 5
  On January $1^\text{st}$ , the temperature in Moscow was - $6 \degree \text{C}$ and the temperature in Rome was $13 \degree \text{C}$ . How many degrees warmer was it in Rome than in Moscow?
  $\begin{aligned} \text{Difference} &= 13 - (- 6 ) \\\\ &= 13 + 6 \\\\ &= 19 \degree \text{C} \end{aligned}$
  Answer
  $19 \degree \text{C}$
6. 6
  The London Eye has $32$ passenger capsules, each of which can take up to $18$ people. What is the maximum number of passengers that can travel at any one time?
  
  $\begin{aligned} \text{Maximum passengers} &= 18 \times 32 \\\\ &= 576 \end{aligned}$
  Answer
  $576$
7. 7
  Find:
  7.a
  $\dfrac{3}{7}\times\dfrac{1}{2}$
  $\begin{aligned} &= \dfrac {3} {7} \times \dfrac {1} {2} \\\\ &= \dfrac {3} {7 \times 2} \\\\ &= \dfrac {3} {14} \end{aligned}$
  Answer
  $\dfrac {3} {14}$
  7.b
  $\dfrac{3}{7}\div\dfrac{1}{2}$
  Keep in mind
  Dividing fractions is the same as multiplying by the reciprocal (inverse) of the second fraction.
  $\begin{aligned} &= \dfrac {3} {7} \div \dfrac {1} {2} \\\\ &= \dfrac {3} {7} \times 2 \\\\ &= \dfrac {3 \times 2} {7} \\\\ &= \dfrac {6} {7} \end{aligned}$
  Answer
  $\dfrac {6} {7}$
8. 8
  What is $4\dfrac{1}{4}-2\dfrac{1}{2}$ ?
  $\begin{aligned} &= 4 \dfrac {1} {4} - 2 \dfrac {1} {2} \\\\ &= \dfrac {17} {4} - \dfrac {5} {2} \\\\ &= \dfrac {17} {4} - \dfrac {5 \times 2} {2 \times 2} \\\\ &= \dfrac {17} {4} - \dfrac {10} {4} \\\\ &= \dfrac {17 - 10} {4} \\\\ &= \dfrac {7} {4} \\\\ &= 1 \dfrac {3} {4} \end{aligned}$
  Answer
  $1 \dfrac {3} {4}$
9. 9
  Calculate the following:
  
  $5 - (2 - 5)^2$
  Keep in mind
  Using BIDMAS
  $\begin{aligned} &= 5 - (2 - 5)^2 \\\\ &= 5 - (-3)^{2} \\\\ &= 5 - 9 \\\\ &= -4 \end{aligned}$
  Answer
  $-4$
10. 10
  Josh has to be at work at $8{:}15 \ \text{am}$ . It takes him $15 \ \text{minutes}$ to get dressed, $20 \ \text{minutes}$ to eat breakfast and $12 \ \text{minutes}$ to drive to work. What is the latest time Josh could get up to get to work on time?
  $\begin{aligned} \text{Time needed} &= 15 + 20 + 12 \\\\ &= 47 \ \text{minutes} \\\\ \text{Get up time} &= 8{:}15 \ \text{am} - 47 \ \text{minutes} \\\\ &= 7{:}28 \ \text{am} \end{aligned}$
  Answer
  $7{:}28 \ \text{am}$
11. 11
  Edward drives to his aunt's house at an average speed of $50 \ \text{km/h}$ , and it takes him $2 \ \text{hours}$ to get there.
  11.a
  How far away does Edward's aunt live?
  Formulas used
  $\text {Distance} = \text{Speed} \times \text{Time}$
  $\begin{aligned} \text{Distance} &= 50 \times 2 \\\\ &= 100 \text{ km} \end{aligned}$
  Answer
  $100 \text{ km}$
  11.b
  What was his average speed home again if it takes him $2\dfrac{1}{2} \ \text{hours}$ to get back?
  Keep in mind
  Dividing fractions is the same as multiplying by the reciprocal (inverse) of the second fraction.
  Formulas used
  $\text {Speed} = \dfrac {\text{Total distance}} {\text{Total time}}$
  $\begin{aligned} \text{Average speed} &= 100 \div 2\dfrac{1}{2} \\\\ &= 100 \div \dfrac{5}{2} \\\\ &= 100 \times \dfrac{2}{5} \\\\ &= \dfrac{100 \times 2}{5} \\\\ &= \dfrac{20 \times 5 \times 2}{5} \\\\ &= \dfrac{20 \times \cancel{5} \times 2}{\cancel{5}} \\\\ &= 20 \times 2 \\\\ &= 40 \text{ km/h} \end{aligned}$
  Answer
  $40 \text{ km/h}$
12. 12
  Write down the next two numbers in the sequence:
  
  $1, \quad -2, \quad 4, \quad -8, \quad \ldots,\quad \ldots$
  Look at how the terms change.
  Multiply by $-2$ each time.
  
  Pattern
  $\begin{matrix} & \rtqMathsSequenceStepBare{\times (-2)} && \rtqMathsSequenceStepBare{\times (-2)} && \rtqMathsSequenceStepBare{\times (-2)} && \rtqMathsSequenceStepBare{\times (-2)} && \rtqMathsSequenceStepBare{\times (-2)} \\ 1, && -2, && 4, && -8, && \ldots, && \ldots \end{matrix}$
  
  Missing numbers
  $\begin{aligned} \text{First missing number} &= -8 \times -2 \\\\ &= 16 \\\\ \text{Second missing number} &= 16 \times -2 \\\\ &= -32 \end{aligned}$
  Answer
  $16$
  $-32$
13. 13
  13.a
  If the shape to the right is a square, work out the value of $x$ .
  
  From the question,
  
  $\begin{aligned} 2x + 5 &= 21 \\\\ 2x &= 21 - 5 \\\\ 2x &= 16 \\\\ x &= \dfrac {16} {2} \\\\ x &= \dfrac {\cancel{2} \times 8} {\cancel{2}} \\\\ x &= 8 \end{aligned}$
  Answer
  $8$
  13.b
  A regular pentagon has the same side length as the square. What is its perimeter?
  Formulas used
  $\text{Perimeter of regular polygon} = \text{n} \times \text{side}$
  $\begin{aligned} \text{Perimeter} &= 5 \times 21 \\\\ &= 105 \ \text{cm} \end{aligned}$
  Answer
  $105 \ \text{cm}$
14. 14
  It takes Karen $40 \ \text{minutes}$ to get to school in the morning. The pie chart shows how the time is divided.
  
  14.a
  How long does Karen spend on the bus?
  $\begin{aligned} \text{Bus time} &= \dfrac {1} {2} \times 40 \\\\ &= \dfrac {40} {2} \\\\ &= \dfrac {2 \times 20} {2} \\\\ &= \dfrac {\cancel 2 \times 20} { \cancel 2} \\\\ &= 20 \ \text{minutes} \end{aligned}$
  Answer
  $20 \ \text{minutes}$
  14.b
  How long does Karen spend walking?
  $\begin{aligned} \text{Walk fraction} &= \dfrac {1} {4} + \dfrac {1} {8} \\\\ &= \dfrac {2} {8} + \dfrac {1} {8} \\\\ &= \dfrac {3} {8} \\\\ \text{Walking time} &= \dfrac {3} {8} \times 40 \\\\ &= \dfrac {3 \times 40} {8} \\\\ &= \dfrac {3 \times 8 \times 5} {8} \\\\ &= \dfrac {3 \times \cancel 8 \times 5} {\cancel 8} \\\\ &= 15 \ \text{minutes} \end{aligned}$
  Answer
  $15 \ \text{minutes}$
15. 15
  Calculate the value of $x$ in the triangle shown.
  
  Formulas used
  $\text{Sum of angles in a triangle} = 180 \degree$
  In the diagram,
  
  $\begin{aligned} 80 + 60 + x &= 180 \\\\ x &= 180 - 80 - 60 \\\\ x &= 40 \degree \end{aligned}$
  Answer
  $40 \degree$
16. 16
  The plan on the right shows a garden. There is a $2 \ \text{m}$ wide path around the edge of the garden, with a swimming pool inside the path. Find the area of the path.
  
  Formulas used
  $\text{Area of rectangle} = \text{length} \times \text{width}$
  Plan
  Find the garden area, find the pool area, then subtract to get the path area.
  
  Garden area
  $\begin{aligned} \text{Garden area} &= 15 \times 12 \\\\ &= 180 \ \text{m}^{2} \end{aligned}$
  
  Pool area
  $\begin{aligned} \text{Pool length} &= 15 - 2 - 2 \\\\ &= 11 \ \text{m} \\\\ \text{Pool width} &= 12 - 2 - 2 \\\\ &= 8 \ \text{m} \\\\ \text{Pool area} &= 11 \times 8 \\\\ &= 88 \ \text{m}^{2} \end{aligned}$
  
  Path area
  $\begin{aligned} \text{Path area} &= 180 - 88 \\\\ &= 92 \ \text{m}^{2} \end{aligned}$
  Answer
  $92 \ \text{m}^{2}$
17. 17
  Gary and Martin have some money. Martin has $\pounds 13$ more than Gary, together they have $\pounds 51$ . How much money does Martin have?
  Define variables
  Let the money Gary has $= a$
  Money Martin has $= 13 + a$
  
  Equation
  From the question,
  $\begin{aligned} a + 13 + a &= 51 \\\\ 2a &= 51 - 13 \\\\ 2a &= 38 \\\\ a &= \dfrac {38} {2} \\\\ a &= \dfrac {2 \times 19} {2} \\\\ a &= \dfrac {\cancel{2} \times 19} {\cancel{2}} \\\\ a &= \pounds 19 \end{aligned}$
  
  Answer
  $\begin{aligned} \text{Money Gary has} &= \pounds 19 \\\\ \text{Money Martin has} &= 19 + 13 \\\\ &= \pounds 32 \end{aligned}$
  Answer
  $\pounds 32$
18. 18
  In the diagram below, the point $B$ has coordinates $(1,4)$ .
  
  18.a
  Write down the coordinates of point $A$ .
  $(2, 1)$
  Answer
  $(2, 1)$
  18.b
  The point $C$ has coordinates $(7,1)$ . Mark $C$ on the diagram and then draw a line from $B$ to $C$ .
  Graph
  18.c
  The point $D$ is on the line you have drawn, and it is twice as far from $B$ as from $C$ . Mark $D$ on the diagram and write down its coordinates.
  Graph
  
  Coordinates
  From the graph,
  $\begin{aligned} B &= (1,4) \\\\ C &= (7,1) \\\\ \text{x-coordinate} &= 1 + \dfrac {2} {3} \times (7 - 1) \\\\ &= 1 + \dfrac {2 \times 6} {3} \\\\ &= 1 + \dfrac {2 \times 2 \times 3} {3} \\\\ &= 1 + \dfrac {2 \times 2 \times \cancel{3}} {\cancel{3}} \\\\ &= 1 + 4 \\\\ &= 5 \\\\ \text{y-coordinate} &= 4 + \dfrac {2} {3} \times (1 - 4) \\\\ &= 4 - \dfrac {2 \times 3} {3} \\\\ &= 4 - \dfrac {2 \times \cancel{3}} {\cancel{3}} \\\\ &= 4 - 2 \\\\ &= 2 \\\\ D &= (5,\ 2) \end{aligned}$
  Answer
  $(5, 2)$
19. 19
  In a car park there are $60$ cars. $\dfrac {5}{12}$ of the cars are red and $20\%$ of the cars are blue. How many cars are there that are neither red nor blue?
  $\begin{aligned} \text{Red cars} &= \dfrac {5} {12} \times 60 \\\\ &= \dfrac {5 \times 60} {12} \\\\ &= \dfrac {5 \times 5 \times 12} {12} \\\\ &= \dfrac {5 \times 5 \times \cancel {12}} {\cancel {12}} \\\\ &= 25 \\\\ \text{Blue cars} &= 20\% \times 60 \\\\ &= \dfrac {20 \times 60} {100} \\\\ &= \dfrac {20 \times 12 \times 5} {20 \times 5} \\\\ &= \dfrac {\cancel {20} \times 12 \times \cancel {5}} {\cancel {20} \times \cancel {5}} \\\\ &= 12 \\\\ \text{Neither} &= 60 - 25 - 12 \\\\ &= 23 \end{aligned}$
  Answer
  $23$
20. 20
  There are $5$ competitors in a tennis competition. If each player plays every other player once only, how many matches will there be?
  Define variables
  Let the $5$ competitors $= A, B, C, D, E$
  Ensure count each match correctly, so $A$ vs $B$ and $B$ vs $A$ are not counted twice.
  
  Matches with A
  $A$ plays:
  
  $A$ vs $B$
  
  $A$ vs $C$
  
  $A$ vs $D$
  
  $A$ vs $E$
  
  Count $= 4$
  
  Matches with B
  $B$ plays:
  
  $B$ vs $C$
  
  $B$ vs $D$
  
  $B$ vs $E$
  
  Count $= 3$
  
  Matches with C
  $C$ plays:
  
  $C$ vs $D$
  
  $C$ vs $E$
  
  Count $= 2$
  
  Matches with D
  $D$ plays:
  
  $D$ vs $E$
  
  Count $= 1$
  
  Count from the list
  $\begin{aligned} \text{Total matches} &= 4 + 3 + 2 + 1 \\\\ &= 10 \end{aligned}$
  Answer
  $10$
21. 21
  A new mathematical operation has been invented. For any two numbers $x \rtqMathsSymbolBoxDot y$ means multiply $x$ by three, then add $y$ , so $4 \rtqMathsSymbolBoxDot 2$ means $4 \times 3 + 2 = 12 + 2 = 14$ .
  21.a
  What is $6 \rtqMathsSymbolBoxDot 4$ ?
  $\begin{aligned} &= 6 \rtqMathsSymbolBoxDot 4 \\\\ &= 6 \times 3 + 4 \\\\ &= 18 + 4 \\\\ &= 22 \end{aligned}$
  Answer
  $22$
  21.b
  What values of $a$ makes $a \rtqMathsSymbolBoxDot 5 = 29$ ?
  From the question,
  
  $\begin{aligned} a \rtqMathsSymbolBoxDot 5 &= 29 \\\\ a \times 3 + 5 &= 29 \\\\ 3a &= 29 - 5 \\\\ 3a &= 24 \\\\ a &= \dfrac{24}{3} \\\\ a &= \dfrac{8 \times 3}{3} \\\\ a &= \dfrac{8 \times \cancel{3}}{\cancel{3}} \\\\ a &= 8 \end{aligned}$
  Answer
  $8$
  21.c
  Find $b$ if $b \rtqMathsSymbolBoxDot b = 52$ .
  From the question,
  
  $\begin{aligned} b \rtqMathsSymbolBoxDot b &= 52 \\\\ b \times 3 + b &= 52 \\\\ 3b + b &= 52 \\\\ 4b &= 52 \\\\ b &= \dfrac{52}{4} \\\\ b &= \dfrac{13 \times \cancel{4}}{\cancel{4}} \\\\ b &= 13 \end{aligned}$
  Answer
  $13$
22. 22
  The bar graph below shows the number of television sets in each house in a street.
  
  22.a
  How many houses have $2$ televisions?
  $11$
  Answer
  $11$
  22.b
  How many houses are there in the street?
  $\begin{aligned} \text{Houses on the street} &= 13 + 11 + 5 + 1 \\\\ &= 30 \end{aligned}$
  Answer
  $30$
  22.c
  How many television sets are there in the street?
  $\begin{aligned} \text{TV sets} &= 13 \times 1 + 11 \times 2 + 5 \times 3 + 1 \times 4 \\\\ &= 13 + 22 + 15 + 4 \\\\ &= 54 \end{aligned}$
  Answer
  $54$
23. 23
  At a birthday party, one half drank only lemonade, one third drank only cola, $15$ people drank neither, and nobody drank both. How many people were at the party?
  Define variables
  Let the total number of people $= a$
  People who only had lemonade $= \dfrac {a} {2}$
  People who only had cola $= \dfrac {a} {3}$
  People who had neither $= 15$
  
  Equation
  From the question,
  $\begin{aligned} \dfrac {a} {2} + \dfrac {a} {3} + 15 &= a \\\\ a - \dfrac {a} {2} - \dfrac {a} {3} &= 15 \\\\ \dfrac {a \times 6} {6} - \dfrac {a \times 3} {2 \times 3} - \dfrac {a \times 2} {3 \times 2} &= 15 \\\\ \dfrac {6a} {6} - \dfrac {3a} {6} - \dfrac {2a} {6} &= 15 \\\\ \dfrac {6a - 3a - 2a} {6} &= 15 \\\\ \dfrac {a} {6} &= 15 \\\\ a &= 15 \times 6 \\\\ a &= 90 \end{aligned}$
  
  Answer
  $\begin{aligned} \text{Total number of people} &= 90 \end{aligned}$
  Answer
  $90$

Reigate Grammar School Paper 1

Questions

Define variables

Side length

Answer

Pattern

Missing numbers

Plan

Garden area

Pool area

Path area

Define variables

Equation

Answer

Graph

Graph

Coordinates

Define variables

Matches with A

Matches with B

Matches with C

Matches with D

Count from the list

Define variables

Equation

Answer