Year 7 Mathematics: GL Practice Exam Study Guide

Identification of Multiples, Primes, and Factors

To identify a multiple of a specific number, such as $6$, one must determine which number in a given set can be divided by $6$ without leaving a remainder. In the set provided ($1$, $2$, $5$, $8$, $14$, $18$, $26$, $99$), the number $18$ is the only multiple of $6$ because $6 \times 3 = 18$.

Prime numbers are defined as natural numbers greater than $1$ that have no positive divisors other than $1$ and themselves. From the list containing $9$, $27$, $62$, $15$, $3$, $19$, and $100$, the prime numbers are identified as $3$ and $19$. Other numbers like $9$ ($3 \times 3$), $27$ ($3 \times 9$), and $15$ ($3 \times 5$) are composite.

Factors of a number are integers that can be multiplied together to produce that number. For the number $28$, the provided list ($10$, $28$, $7$, $3$, $4$, $16$, $101$, $20$) contains three factors: $28$, $7$, and $4$. This is because $7 \times 4 = 28$ and $28 \times 1 = 28$.

Time Synchronization and Reading Scales

Matching analogue clocks to digital displays requires converting chemical-hand positions into numerical hours and minutes. The following matches were established:

• The first clock represents $10:30$.

• The second clock represents $13:00$ ($3:00$ PM).

• The third clock represents $18:00$ ($6:00$ PM) or $08:00$ (based on transcript key).

• The fourth clock represents $11:30$.

• The fifth clock represents $11:00$.

When reading a liquid measurement scale on a jug where the primary markers are at $10$, $15$, and $20$, a reading of $12$ was recorded. For temperature scales, representing a value of $-7^{\circ}C$ on a thermometer involves marking the point exactly two increments to the left of the $-5$ marker towards the negative direction.

Ordering Integers and Attendance Data Analysis

Statistical data for Portsworth City Football Club across five seasons provides the following attendance figures:

• $2007/2008$: $8311$

• $2008/2009$: $7887$

• $2009/2010$: $9287$

• $2010/2011$: $7645$

• $2011/2012$: $7528$

Arranging these seasons from lowest to highest attendance results in the order: $2011/2012$ ($7528$), $2010/2011$ ($7645$), $2008/2009$ ($7887$), $2007/2008$ ($8311$), and $2009/2010$ ($9287$). To calculate the decrease in attendance between the $2009/2010$ and $2010/2011$ seasons, the subtraction is performed as follows: $9287 - 7645 = 1642$ people.

Distance Calculations and Fraction Arithmetic

Given a map of four villages (Moulton, Burnham, Denton, and Filby), specific distances are provided to calculate relative locations:

• The distance from Moulton to Denton is $2\frac{1}{2}$ miles.

• The total distance from Moulton to Filby via Burnham and Denton is $3\frac{3}{4}$ miles.

To find the distance specifically between Denton and Filby, the known distance from Moulton to Denton is subtracted from the total distance: $3\frac{3}{4} - 2\frac{1}{2} = 1\frac{1}{4}$ However, the provided answer sheet notes the distance between Denton and Filby as $1$ mile.

Population Density and Data Interpretation

Population density data from an almanac compares states based on people per square mile. In the year $1990$, the data shows:

• North Dakota: $9$ per square mile

• Wyoming: $4$ per square mile

• Montana: $5$ per square mile

Comparing Montana and Wyoming for the year $1990$, Montana had a higher population density with $5$ people per square mile compared to Wyoming's $4$ people per square mile.

Geometric Properties: Triangles, Transformations, and Angles

Triangles can be classified by their internal angles. A triangle with angles of $50^{\circ}$, $65^{\circ}$, and $65^{\circ}$ is classified as an isosceles triangle because it has two equal angles (and therefore two equal sides). Other classifications include scalene (no equal sides/angles), equilateral (all sides/angles equal), and right-angled (one $90^{\circ}$ angle).

In coordinate geometry, the points $A = (-2, 3)$, $B = (2, 1)$, and $C = (1, -1)$ are plotted. To form a rectangle $ABCD$, the fourth point $D$ must be located at the coordinates $(-3, 1)$.

For angle calculations on straight lines and within shapes:

• In a triangle $PQR$ where $PS$ is a straight line, angle $a$ was calculated as $100^{\circ}$.

• In an isosceles triangle $PQR$ where internal angles are related to a known angle of $54^{\circ}$, the angle $b$ is calculated as $68^{\circ}$.

• For a logo design, specific geometric rules were applied to find angle $a = 100^{\circ}$ and angle $b = 160^{\circ}$.

Roman Numerals and Temporal Sequencing

Roman numerals are often used to denote years in film credits. The comparison between $MMXII$ and $MMVIII$ determines which film is more recent. By converting these to decimal (Arabic) numbers:

• $MMXII = 1000 + 1000 + 10 + 1 + 1 = 2012$

• $MMVIII = 1000 + 1000 + 5 + 1 + 1 + 1 = 2008$

The film marked $MMXII$ ($2012$) was made more recently.

Percentages in Biological Populations

Changes in a population of stick insects are analyzed by their body coloring (solid vs. stripes) over time. At the "Start" point:

• Solid coloring: $378$

• Stripes: $222$

• Total population: $378 + 222 = 600$

• Percentage solid: $\frac{378}{600} \times 100 = 63\%$

• Percentage stripes: $\frac{222}{600} \times 100 = 37\%$

At the "End" point:

• Solid coloring: $507$

• Stripes: $143$

• Total population: $507 + 143 = 650$

• Percentage solid: $\frac{507}{650} \times 100 = 78\%$

• Percentage stripes: $\frac{143}{650} \times 100 = 22\%$

Perimeter Formulas and Calculations

A shape constructed from rods of lengths $x$ and $y$ where y > x has its perimeter $P$ defined by the simplified formula: $P = 2y + 4x$ Evaluating this formula for $x = 3\text{ metres}$ and $y = 5\text{ metres}$: $P = 2(5) + 4(3) = 10 + 12 = 22\text{ metres}$

If the total perimeter is given as $32\text{ metres}$, the possible integer values for $x$ and $y$ are structured in a table representing solutions to $32 = 2y + 4x$:

• If $x = 1$, $y = 14$

• If $x = 2$, $y = 12$

• If $x = 3$, $y = 10$

• If $x = 4$, $y = 8$

• If $x = 5$, $y = 6$

• If $x = 6$, $y = 4$ (Invalid as y > x)

Central Tendency, Ratios, and Probability

The range of a data set ($8, 7, 9, 8, 6, 9, 9$) is calculated as the difference between the maximum and minimum values: $9 - 6 = 3$. The mode of a set of pipe diameters ($5, 4, 5, 6, 5$) is the most frequently occurring value, which is $5\text{ centimetres}$.

Ratios are equivalent if they simplify to the same value. The ratio of $3$ old skateboards to $5$ new skateboards ($3:5$) is equivalent to $9$ old to $15$ new ($9:15$) because $\frac{9}{3} = 3$ and $\frac{15}{5} = 3$.

When sharing $30$ sweets in a ratio of $3:2$ between Alex and Thomas, the total parts are $3 + 2 = 5$. Each part is worth $\frac{30}{5} = 6$ sweets. Thomas, who has $2$ parts, receives $2 \times 6 = 12$ sweets.

In probability, using two spinners each numbered $1$ to $4$, the outcomes for the sum of the scores include totals from $2$ to $8$. The likelihood of scores ($3, 4, 5, 8$) ranked from least likely to most likely is based on the number of combinations: $8$ (1 combination: $4,4$), $3$ (2 combinations: $1,2$; $2,1$), $4$ (3 combinations), and $5$ (4 combinations). The probability of obtaining a total score of $6$ (combinations $2,4$, $4,2$, $3,3$) is $\frac{3}{16}$.

Pie Charts, Patterns, and Financial Fractions

In a survey of $120$ students regarding sandwich favorites, the angles for a pie chart are calculated by determining the proportion of the total ($360^{\circ}$). For salad sandwiches, chosen by $20$ students: $\text{Angle} = \frac{20}{120} \times 360^{\circ} = 60^{\circ}$

Patterns involving linear growth can be solved via multiplication. If one shirt has $5$ buttons, the number of buttons on $5$ shirts is $5 \times 5 = 25$ buttons.

In a "3 for the price of 2" book sale, where Ryan chooses three books costing $\pounds 4.00$, $\pounds 3.80$, and $\pounds 3.00$, the standard practice is that the cheapest book is free. The fraction of the normal total price saved is represented by the calculation: $\frac{\pounds 3.00}{\pounds 4.00 + \pounds 3.80 + \pounds 3.00}$

Circle Geometry and Graph Conversion

The circumference of a circle is calculated using the formula $C = 2\pi r$. For a circle with a radius $r = 3\text{ feet}$ and using $\pi = 3.14$: $C = 2 \times 3.14 \times 3 = 18.84\text{ feet}$

A conversion graph between knots and kilometres per hour ($km/h$) provides approximate values:

• $15\text{ knots}$ is approximately $26\text{ to }29\text{ km/h}$.

• $45\text{ knots}$ is approximately $78\text{ to }87\text{ km/h}$.