Teacher edition (with key)← Advanced Math 7

Advanced Math 7 — every item

Singapore Dimensions Math 7A / 7B · 80 items

One section per unit, in course order. For review and for a year of homework sheets.

01Factors and multiples: Euclid's algorithmeuclid-gcd

1.
Write $360$ in index notation as $2^a \cdot 3^b \cdot 5^c$ . What is $a + b + c$ ?
Answer: ______________
2.
Use Euclid's algorithm to find $\gcd(252, 198)$ , then use $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ . Enter $\operatorname{lcm}(252, 198)$ .
Answer: ______________
3.
A rectangular patio measures $360$ cm by $252$ cm. It is to be covered exactly with identical square tiles, as large as possible, with no cutting. How many tiles are needed?
Answer: ______________
4.
Running Euclid's algorithm on $1071$ and $462$ gives these lines:

$1071 = 2 \times 462 + 147,\qquad 462 = 3 \times 147 + 21,\qquad 147 = 7 \times 21 + 0.$

What is $\gcd(1071, 462)$ ?
1. (A)
  $3$
2. (B)
  $7$
3. (C)
  $21$
4. (D)
  $147$
5.
Two positive integers have a product of $1080$ and a greatest common factor of $6$ . What is their least common multiple?
1. (A)
  $6$
2. (B)
  $36$
3. (C)
  $180$
4. (D)
  $216$
5. (E)
  $1080$

02Real numbersreals-radicals-sci-notation

1.
$\sqrt{50}$ lies between two consecutive whole numbers. Enter the smaller one.
Answer: ______________
2.
Write $4{,}560{,}000$ in scientific notation $a \times 10^k$ with $1 \le a < 10$ . Enter $a$ .
Answer: ______________
3.
Which list contains only rational numbers?
1. (A)
  $\sqrt{50},\ \dfrac{22}{7},\ 0.5$
2. (B)
  $\pi,\ 3.14,\ \dfrac{22}{7}$
3. (C)
  $\sqrt{2},\ \sqrt{4},\ \sqrt{8}$
4. (D)
  $\sqrt{49},\ 0.\overline{3},\ -\dfrac{5}{8}$
4.
Which statement is true?
1. (A)
  $\sqrt{\dfrac{16}{25}}$ is irrational because it is a square root.
2. (B)
  $\pi = \dfrac{22}{7}$ , so $\pi$ is rational.
3. (C)
  $\sqrt{\dfrac{16}{25}} = \dfrac{4}{5}$ , so it is rational.
4. (D)
  $0.\overline{3}$ is irrational because its decimal never ends.
5.
How many integers $n$ satisfy $\sqrt{10} < n < \sqrt{90}$ ?
1. (A)
  $5$
2. (B)
  $6$
3. (C)
  $7$
4. (D)
  $8$
5. (E)
  $80$

03Introduction to algebraalgebra-foundations-7

1.
In the expression $5x^2 - 3x + 7$ , what is the coefficient of $x$ ?
Answer: ______________
2.
Write an expression for “seven less than twice a number $n$ .”
Answer: ______________
3.
Here are four algebraic sentences.

(i) $3x + 2 = 11$ (ii) $A = \pi r^2$ (iii) $2(x + 3) = 2x + 6$ (iv) $4x - 7$

Which one is an identity — true for every value of $x$ ?
1. (A)
  (i)
2. (B)
  (ii)
3. (C)
  (iii)
4. (D)
  (iv)
4.
Which of these is an expression (not an equation)?
1. (A)
  $x = 5$
2. (B)
  $\dfrac{1}{2}x - 4$
3. (C)
  $2x + 3 = 9$
4. (D)
  $y = 3x$
5.
What is the value of $x^3 - 2x^2 + 3x - 4$ when $x = -2$ ?
Answer: ______________

04Algebraic manipulationexpand-factor-7

1.
Simplify $\dfrac{1}{2}x + \dfrac{3}{4}x - 2x$ . Enter the coefficient of $x$ in the result.
Answer: ______________
2.
Expand and simplify $(2x - 3)(x + 5)$ .
Answer: ______________
3.
Factor $xy + 2x + 3y + 6$ completely.
Answer: ______________
4.
Expand $-2(x - 5)$ .
1. (A)
  $-2x - 10$
2. (B)
  $2x - 10$
3. (C)
  $-2x + 10$
4. (D)
  $-2x - 5$
5.
When $(2x + 3)(x - 4)$ is expanded and written as $ax^2 + bx + c$ , what is $a + b + c$ ?
1. (A)
  $-15$
2. (B)
  $-10$
3. (C)
  $-3$
4. (D)
  $1$
5. (E)
  $15$

05Simple equations in one variablelinear-equations-7

1.
Solve $5x - 7 = 3x + 9$ .
Answer: ______________
2.
Solve $\dfrac{x + 5}{3} = \dfrac{x - 1}{2}$ .
Answer: ______________
3.
Maya is three times as old as Ben. In $8$ years she will be twice as old as Ben will be then. How old is Ben now?
Answer: ______________
4.
Solve $\dfrac{x}{3} + 2 = \dfrac{x}{4} + 5$ .
1. (A)
  $3$
2. (B)
  $12$
3. (C)
  $36$
4. (D)
  $84$
5.
The length of a rectangle is $3$ cm more than twice its width. Its perimeter is $48$ cm. What is its area, in cm²?
1. (A)
  $63$
2. (B)
  $105$
3. (C)
  $119$
4. (D)
  $144$
5. (E)
  $238$

06Ratio, rate, and speed (motion graphs)speed-motion-graphs

1.
A train travels at a steady $80$ km/h for $2$ hours $15$ minutes. How far does it travel, in km?
Answer: ______________
2.
Sara drives $60$ km from home to a town at $30$ km/h and returns along the same road at $60$ km/h. What is her average speed for the whole round trip, in km/h?
Answer: ______________
3.
A speed–time graph for a tram: the speed rises in a straight line from $0$ to $20$ m/s during the first $10$ seconds, then stays at $20$ m/s for the next $30$ seconds. How far does the tram travel in these $40$ seconds, in metres?
Answer: ______________
4.
A bus goes from town A to town B at $40$ km/h and returns along the same road at $60$ km/h. What is its average speed for the whole journey?
1. (A)
  $48$ km/h
2. (B)
  $50$ km/h
3. (C)
  $52$ km/h
4. (D)
  $100$ km/h
5.
A walker leaves a village at 8:00 a.m. at a steady $4$ km/h. At 9:30 a.m. a cyclist leaves the same village along the same road at a steady $12$ km/h. At what time does the cyclist catch the walker?
1. (A)
  10:00 a.m.
2. (B)
  10:15 a.m.
3. (C)
  10:30 a.m.
4. (D)
  10:45 a.m.
5. (E)
  11:00 a.m.

07Percentage: reverse percent and simple interestpercent-simple-interest

1.
Find the simple interest earned on $500 at $4\%$ per year for $3$ years, in dollars.
Answer: ______________
2.
After a $20\%$ discount, a jacket costs $72. What was the original price, in dollars?
Answer: ______________
3.
Priya deposits $1,200 in an account paying $5\%$ simple interest per year. After how many years will the account hold $1,500?
Answer: ______________
4.
A video game is on sale for $60 after a $25\%$ discount. What was the original price?
1. (A)
  $45
2. (B)
  $75
3. (C)
  $80
4. (D)
  $85
5.
Account A starts with $800 and earns $6\%$ simple interest per year. Account B starts with $1,000 and earns $4\%$ simple interest per year. After how many years do the two accounts hold the same amount?
1. (A)
  $5$
2. (B)
  $10$
3. (C)
  $20$
4. (D)
  $25$
5. (E)
  $50$

08Angles, triangles, and quadrilateralseuclidean-angles-polygons

1.
Two angles are supplementary. One measures $37^\circ$ . What is the other, in degrees?
Answer: ______________
2.
What is the size of each interior angle of a regular nonagon ( $9$ sides), in degrees?
Answer: ______________
3.
In triangle $ABC$ , side $BC$ is extended past $C$ to a point $D$ . The exterior angle $\angle ACD = 112^\circ$ and $\angle BAC = 47^\circ$ . Find $\angle ABC$ , in degrees.
Answer: ______________
4.
Lines $AB$ and $CD$ are parallel. A transversal meets $AB$ at $P$ and $CD$ at $Q$ . Angles $\angle BPQ$ and $\angle DQP$ lie between the parallel lines on the same side of the transversal (co-interior). If $\angle BPQ = 64^\circ$ , what is $\angle DQP$ ?
1. (A)
  $26^\circ$
2. (B)
  $64^\circ$
3. (C)
  $116^\circ$
4. (D)
  $296^\circ$
5.
The interior angles of a pentagon measure $x^\circ$ , $(x + 10)^\circ$ , $(x + 20)^\circ$ , $(x + 30)^\circ$ , and $(x + 40)^\circ$ . What is the measure of the largest angle, in degrees?
1. (A)
  $92$
2. (B)
  $108$
3. (C)
  $118$
4. (D)
  $128$
5. (E)
  $148$

09Number patternssequences-patterns-7

1.
What is the $10$ th term of the arithmetic sequence $4, 11, 18, 25, \ldots$ ?
Answer: ______________
2.
Write a formula for the $n$ th term $T_n$ of the sequence $7, 12, 17, 22, \ldots$ in terms of $n$ . Give it in the form $an + b$ .
Answer: ______________
3.
A pattern of dots grows as shown. Each figure is a triangle of dots with one more row than the figure before.

Figure 1 2 3 4
Dots 3 6 10 15

How many dots are in Figure $8$ ?
Answer: ______________
4.
Which formula gives the $n$ th term of $3, 7, 11, 15, \ldots$ ?
1. (A)
  $T_n = 4n + 3$
2. (B)
  $T_n = 3n + 4$
3. (C)
  $T_n = n + 4$
4. (D)
  $T_n = 4n - 1$
5.
How many terms of the arithmetic sequence $7, 10, 13, 16, \ldots$ are less than $100$ ?
1. (A)
  $30$
2. (B)
  $31$
3. (C)
  $32$
4. (D)
  $33$
5. (E)
  $34$

Figure	1	2	3	4
Dots	3	6	10	15

10Coordinates and linear graphsslope-linear-graphs

1.
Find the slope of the line through $(2, 5)$ and $(6, 17)$ .
Answer: ______________
2.
Write an equation of the line through $(0, 3)$ and $(4, 1)$ .
Answer: ______________
3.
The water level in a tank, in cm, after $t$ minutes of draining is given by $h = 60 - 4t$ . Which statement is correct?
1. (A)
  The tank starts at $60$ cm and the level falls $4$ cm every minute.
2. (B)
  The tank starts at $4$ cm and the level falls $60$ cm every minute.
3. (C)
  The tank starts at $60$ cm and the level rises $4$ cm every minute.
4. (D)
  The tank is empty after $4$ minutes.
4.
What is the slope of the line through $(1, 2)$ and $(4, 8)$ ?
1. (A)
  $\dfrac{1}{2}$
2. (B)
  $2$
3. (C)
  $-2$
4. (D)
  $6$
5.
The points $(-2, 11)$ , $(1, 2)$ , and $(4, k)$ lie on the same straight line. What is $k$ ?
Answer: ______________

11Inequalities (multi-step)linear-inequalities-7

1.
Solve $3x + 5 < 20$ . Enter the largest integer that satisfies it.
Answer: ______________
2.
Solve $5 - 2x > 11$ . Enter the largest integer that satisfies it.
Answer: ______________
3.
On a number line, a closed circle is drawn at $-1$ , an open circle at $4$ , and the segment between them is shaded. Which inequality is graphed?
1. (A)
  $-1 < x < 4$
2. (B)
  $-1 \le x < 4$
3. (C)
  $-1 < x \le 4$
4. (D)
  $-1 \le x \le 4$
4.
Solve $-2x + 7 > 13$ .
1. (A)
  $x < 3$
2. (B)
  $x > -3$
3. (C)
  $x > 3$
4. (D)
  $x < -3$
5.
How many integers $x$ satisfy $1 \le 7 - 2x < 15$ ?
Answer: ______________

12Perimeters and areas of plane figures (circles)circles-arcs-sectors

1.
A circle has diameter $14$ cm. Its circumference is $k\pi$ cm. Enter $k$ .
Answer: ______________
2.
A sector of a circle has radius $9$ cm and central angle $120^\circ$ . Its area is $k\pi$ cm². Enter $k$ .
Answer: ______________
3.
A square has side $8$ cm. A quarter circle with its center at one corner of the square and radius $8$ cm is cut away. Using $\pi \approx 3.14$ , what is the area of the remaining piece, in cm²?
Answer: ______________
4.
A sector has radius $6$ cm and central angle $60^\circ$ . What is its area?
1. (A)
  $\dfrac{3}{2}\pi$ cm²
2. (B)
  $2\pi$ cm²
3. (C)
  $6\pi$ cm²
4. (D)
  $36\pi$ cm²
5.
A circle is inscribed in a square of side $12$ , touching all four sides. The total area of the four corner regions — inside the square but outside the circle — can be written as $a - b\pi$ for positive integers $a$ and $b$ . What is $a + b$ ?
1. (A)
  $108$
2. (B)
  $144$
3. (C)
  $156$
4. (D)
  $180$
5. (E)
  $288$

13Volume and surface area of solidsvolume-surface-3d

1.
A cylinder has radius $3$ cm and height $10$ cm. Its volume is $k\pi$ cm³. Enter $k$ .
Answer: ______________
2.
A cone has radius $6$ cm and height $8$ cm. Its total surface area is $k\pi$ cm². Enter $k$ .
Answer: ______________
3.
A grain silo is a cylinder of radius $3$ m and height $10$ m, topped by a hemisphere of radius $3$ m. The volume of the silo is $k\pi$ m³. Enter $k$ .
Answer: ______________
4.
A square pyramid has a base edge of $6$ cm and a height of $5$ cm. What is its volume?
1. (A)
  $10$ cm³
2. (B)
  $60$ cm³
3. (C)
  $90$ cm³
4. (D)
  $180$ cm³
5.
A cylindrical jug of radius $4$ cm is filled with water to a height of $9$ cm. All the water is poured into an empty cone-shaped glass of radius $6$ cm, which it exactly fills. What is the height of the cone, in cm?
1. (A)
  $4$
2. (B)
  $6$
3. (C)
  $9$
4. (D)
  $12$
5. (E)
  $36$

14Proportionsdirect-inverse-proportion

1.
$y$ is directly proportional to $x$ , and $y = 12$ when $x = 4$ . Find $y$ when $x = 7$ .
Answer: ______________
2.
The table shows pairs of values of $x$ and $y$ . Decide whether $y$ is directly or inversely proportional to $x$ , then find the missing value.

$x$ $2$ $4$ $10$
$y$ $30$ $15$ $?$

Enter the missing value of $y$ .
Answer: ______________
3.
A spring stretches $6$ cm when a $1.5$ kg mass hangs from it and $14$ cm when a $3.5$ kg mass hangs from it. The stretch $y$ (in cm) is directly proportional to the mass $x$ (in kg). Write the rule for $y$ in terms of $x$ .
Answer: ______________
4.
It takes $6$ painters $10$ days to paint a fence. Working at the same rate, how many days would it take $12$ painters?
1. (A)
  $2.5$ days
2. (B)
  $5$ days
3. (C)
  $20$ days
4. (D)
  $60$ days
5.
$a$ is directly proportional to $b$ , and $b$ is inversely proportional to $c$ . When $c = 4$ , $a = 6$ . What is $a$ when $c = 12$ ?
Answer: ______________

$x$	$2$	$4$	$10$
$y$	$30$	$15$	$?$

15Data handlingdata-handling-7

1.
Which of these is continuous data?
1. (A)
  The heights of students, in cm
2. (B)
  The number of students in each class
3. (C)
  The shoe sizes sold in a shop
4. (D)
  The number of goals scored in each match
2.
A back-to-back stem-and-leaf plot shows test scores for two classes. Leaves are read outward from the stem, so the top row means Class A scored $58$ and $55$ , and Class B scored $52$ and $56$ .

Class A leaves Stem Class B leaves
8 5 5 2 6
7 3 1 6 0 4 4 8
6 2 7 1 5 9
0 8 3

Find the median of Class B minus the median of Class A.
Answer: ______________
3.
Eight students recorded their hours of sleep on Saturday night: $6, 7, 7, 7, 8, 8, 9, 20$ . Find the mean minus the median.
Answer: ______________
4.
The table shows how long $20$ students spent on homework.

Minutes 0–10 10–20 20–30
Students $4$ $10$ $6$

Estimate the mean time, in minutes.
1. (A)
  $10$
2. (B)
  $15$
3. (C)
  $16$
4. (D)
  $20$
5.
The mean of five numbers is $12$ . When one of the numbers is removed, the mean of the remaining four is $10$ . What number was removed?
1. (A)
  $2$
2. (B)
  $10$
3. (C)
  $12$
4. (D)
  $20$
5. (E)
  $22$

Class A leaves	Stem	Class B leaves
8 5	5	2 6
7 3 1	6	0 4 4 8
6 2	7	1 5 9
0	8	3

Minutes	0–10	10–20	20–30
Students	$4$	$10$	$6$

16Probabilityprobability-trees

1.
A bag holds $3$ red, $2$ blue, and $5$ green marbles. One marble is drawn at random. What is the probability that it is not green? Give a fraction in lowest terms.
Answer: ______________
2.
A bag holds $4$ red and $2$ blue counters. Two counters are drawn one after the other without replacement. What is the probability that both are the same colour? Give a fraction in lowest terms.
Answer: ______________
3.
Two fair six-sided dice are rolled. What is the probability that the two numbers add to $8$ ? Give a fraction in lowest terms.
Answer: ______________
4.
Two fair coins are tossed, one after the other. What is the probability of getting heads on the first coin and tails on the second?
1. (A)
  $\dfrac{1}{4}$
2. (B)
  $\dfrac{1}{2}$
3. (C)
  $\dfrac{3}{4}$
4. (D)
  $1$
5.
A bag holds $2$ red and $3$ blue marbles. Two marbles are drawn at random without replacement. What is the probability that at least one of them is red?
1. (A)
  $\dfrac{3}{10}$
2. (B)
  $\dfrac{2}{5}$
3. (C)
  $\dfrac{3}{5}$
4. (D)
  $\dfrac{7}{10}$
5. (E)
  $\dfrac{4}{5}$