TIMO Primary 4 Solman - 1 PDF [PDF]

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Primary 4 Logical Thinking 1) There are 87 ducks and cows on a farm. If there are 242 legs in total, how many more ducks are there than cows? Answer: 19 Solution: We form two equations for the given problem, an equation relating the number of animals and the equation relating the number of legs. Setting D to represent the number of ducks, and C to represent the number of cows, the two equations are: D + C = 87 (this represents that the ducks and chickens combined are 87) 2D + 4C = 242 (this represents that the total number of legs of all the ducks and the total number of legs of all the cows) Using substitution or elimination methods (methods solving two equations with two unknowns), D is known to be 53, C is known to be 34. The difference of the ducks and cows, C - D, is now given as 53 - 34 = 19 2) Kris, Mary, and Genesis have a total of 217 marbles. If Kris has twice the number of marbles Mary has, and Genesis has 7 less than what Mary has, how many marbles does Mary have? Answer: 56 Solution: This can be down easily by writing it down as an equation Kris + Mary + Genesis = 217 Both Kris and Genesis’s marbles are referenced to Mary, so it would be easier to set Mary as X. Since Kris has twice the number of marbles Mary has, Kris has 2X, and since Genesis has 7 marbles less than Mary, Genesis has X - 7. Rewriting the equation, it becomes 2X + X + (X - 7) = 217 4X - 7 = 217 4X = 224 X = 56 3) A pattern is shown below, determine the 57th term in the pattern

□ △ ○ □ △ □ △ ○ □ △ □ △ ○ □ △... Answer: Triangle Solution: The pattern follows the sequence: Square-Triangle-Circle-Square-Triangle and then repeats, in other words, the pattern repeats a sequence of 5 shapes. Dividing 57 by 5 will yield us 11 remainder 2. Taking the remainder and counting from the left, the second one is a circle (square-​triangle​-...) thus the 57th term is a triangle

4) If today is a Friday, what day of the week will 676 days later be? Answer: Tuesday Solution: After friday, the pattern of the days of the week will continue as Sat-Sun-Mon-Tue-Wed-Thurs-Fri and repeats, in other words, it will continue on a sequence of 7 days. Dividing 676 by 7 will give us 96 remainder 4. Taking the remainder and counting from the left, the fourth one is on Tuesday (Sat-Sun-Mon-​Tue​…) thus the 676th day is on a Tuesday 5) Eve, Joe, and Marie are playing a game of stepping tiles. If Eve steps on every 2nd tile, Joe steps on every 3rd tile, and Marie steps on every 11th tile, determine the number of the second tile that will be stepped on by all three friends. Answer: 132 Solution: Eve steps on every second tile, therefore she steps on tiles (2,4,6,8,10,12…). Joe steps on the tiles (3,6,9,12,15,18…) and Marie steps on the tiles (11,22,33, 44,55…) SInce they are all prime, the LCD is 2 x 3 x 11 = 66. Therefore, the first tile they will step on is the 66th, the second one would be 66 x 2 = 132 6) It takes 15 minutes to cut a long piece of log into 6 sections. If the time to cut each section is the same, how many minutes would it take to cut a log into 45 sections? Answer: 132 Solution: It takes 15 mins to cut a log into 6 sections, 6 sections requires 5 cuts, (following sections = cuts + 1), in other words, it takes 15 mins to make 5 cuts, or 3 mins per cut. To create 45 sections, (or 44 cuts), it would take 44 x 3 mins = 132 mins Arithmetic 7) Find the value of 4567 + 324 + 200 - 1567 + 676 + 700 Answer: 4900 Solution: By rearranging the terms, the equation becomes easier to solve = 4567 + 324 + 200 - 1567 + 676 + 700 = (4567 - 1567) + (324 + 676) + (200 + 700) = 3000 + 1000 + 900 = 4900 8) Determine the sum of 1 2 + 2 2 + 3 2 + 4 2 + ... +

230 2

Answer: 4082155 Solution: For the sum of the squares of the first n counting integers, it is given to be (n) x(n + 1) x (2n + 1) /6 Therefore, for n = 230 = (230) x (231) x (461) / 6 = 4082155 9) Determine the value of 222 x 1110 ÷ 37 Answer: 6660 Solution: We can dissect and rearrange the given terms to form an easier set to evaluate = 222 x 1110 ÷ 37 = 10 x 222 x 111 ÷ 37 = 10 x 222 x 3 x 37 ÷37 = 10 x 222 x 3 = 6660 10) Simplify Answer:

143 169

as a fraction in lowest terms

11 13

Solution: Both the numerators and denominators of the fraction can be factored, thus 143/169 can be expressed as (13 x 11)/(13 x 13) Since both numerator and denominator have a factor of 13, we can remove that from both sides, thus, the remaining value is 11/13 11) Evaluate 1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + 17 - 19 + 21 Answer: 11 Solution: The number sentence can be rearranged, forming = 21 - 19 + 17 - 15 + 13 - 11 + 9 - 7 + 5 - 3 + 1 By grouping, = (21 - 19) + (17 - 15) + (13 - 11) + (9 - 7) + (5 - 3) + 1 =2+2+2+2+2+1 = 11 12) Find the value of 9540 ÷ 9 + 9540 ÷ 106 + 9540 ÷ 5 Answer: 2 Solution:Because of the PEMDAS/GEMDAS rule, we are not allowed to directly allowed to evaluate from left to right, the simplest way would be to factor out the similar terms first. = 9540 ÷ 9 + 9540 ÷ 106 + 9540 ÷ 5

= 9540 ÷ (9 x 106 x 5) = (18 x 106 x 5) ÷ (9 x 106 x 5) By canceling out similar terms, = 18 ÷ 9 =2

Number Theory 13) GIven that ​456A22​is a six-digit number that is divisible by 9. Determine the value of A Answer: 8 Solution: For a number to be divisible by 9, its digits should sum up to a number divisible by 9. In this case, the sum of the digits is 4 + 5 + 6 + A + 2 + 2 = 19 + A The next number after 19 that is divisible by 9 is 27. Thus the value of A should be a complement of 19 that forms 27. Thus, A = 27 - 19 = 8 14) Determine the smallest four-digit number divisible by 19 and 3 Answer: 1026 Solution: Applying the ceiling value (rounded-up value) of the quotient of 1000 (the smallest four-digit number) and the LCD of 19 and 3. Since 19 and 3 are both prime, then their LCD is equal to 19 x 3 = 57 ⎡ 1000 57 ⎤ = ⎡17.54...⎤ = 18 Multiplying the ceiling value with the LCD, 18 x 57 = 1026 15) Determine the last digit of N if N = 2019 2020 Answer: 1 Solution: We exclude all the digits of the base except the digits place. The last digit of the base is 9, when multiplied by 9, becomes (9 x 9) = 81, which ends with 1, when that last digit is multiplied again by 9, it becomes (1 x 9) = 9, when multiplied by 9 again,becomes (9 x 9) = 81, which ends with 1 If the same process is repeated over and over again, it can be seen that the last digit (starting from not being multiplied yet) follows the pattern 9-1-9-1-9-1… Thus, the last digit should be either of the two numbers 9 or 1. SInce, for every odd power, it falls on 9, and for every even power, it falls on 1, for 2020, which is even, should fall on 1

16) Two numbers, A and B, have a sum of 299. If A is 1 less than 11 times B, determine the value of A Answer: 274 Solution: We can set both conditions as two equations, A + B = 299, 11B = A + 1 By substitution, 11B - 1 + B = 299 12B = 300 B = 25 Therefore, since B = 25 A + B = 299 A + 25 = 299 A = 274 17) Determine the value of N if N = 11 x 12 x 13 x 14 x 15 x 16 Answer: 0 Solution: N has a factor of 15, which is a factor of 5. N is also a factor of 12, which is a factor of 12, which is a factor of 2. Since N has factors 2 and 5, it can be concluded that N is a factor of 10. By definition, any factor of 10 should end with a zero, therefore the last digit should be a 0. 18) Determine the average of the numbers below: 43, 34, 56, 78, 90, 21, 28 Answer: 50 Solution: The average of a set of numbers can be obtained by adding all the numbers in the set and dividing the sum by the quantity of numbers added. The sum is 43 + 34 + 56 + 78 + 90 + 21 + 28 = 350 SInce 7 numbers are added to form the sum, 350 /7 = 50 Geometry 19) Determine the number of rectangles in the figure below

Answer: 60

Solution: 1

2

3

4

2 3 The length is 4 units long and the width is 3 units wide. For a rectangle, the total number of rectangles can be evaluated by the sum of all the written digits of the length by all sum of all the written digits of the width, thus for this problem: Rectangles = (1 + 2 + 3 + 4) x (1 + 2 + 3) Rectangles = 10 x 6 Rectangles - 60 20) There are 18 points on a plane, where no three of which are collinear. By connecting points, how many lines can be drawn on the plane? Answer:153 lines Solution: The N number of lines that can be drawn with an M number of points can be expressed in the relation N = (M) x (M - 1) / 2 N = (18) x (18 - 1) / 2 N = (18 x 17) / 2 N = 306 / 2 N = 153

21) A square is formed by combining 12 rectangles, with dimensions 6 cm by 8 cm. Determine the perimeter of the formed square Answer: 96 cm Solution: To form a square, the rectangles must be formed similar to what is shown below. It can be rechecked by verifying that the total length and total width are the same. In this case, 8 + 8 + 8 = 6 + 6 + 6 + 6 = 24

The total side of the newly formed square is now 24 cm. SInce the perimeter of a square is defined as 4S, then the perimeter of this figure is 4 x 24 = 96 cm 22) A rectangle has a perimeter of 122, if the sides are integral values, determine the smallest possible area of the rectangle Answer: 60 Solution: A rectangle forms the smaller area the larger the difference of length and width. The dimensions of a rectangle with the largest difference should be

Since the smallest side can not go lower than 1. The area of this rectangle is given by L x W, therefore, the area is 60 x 1 = 60 23) Determine the perimeter of the figure below

Answer: 78 Solution: By analysis of the figure, the following dimensions can be deduced:

Since the perimeter is the sum of all sides, the perimeter for this figure is 18 + 9 + 9 + 9 + 9 + 9 + 3 + 3 + 6 + 6 = 78 units. 24) Determine the sum of interior angles in a pentagon (five-sided polygon) Answer: 540 degrees Solution: The sum of interior angles of a polygon with N sides is given by (N - 2) x 180 Therefore, for a pentagon, S = 5 = (N - 2) x 180 = (5 - 2) x 180 = 540 Combinatorics 25) How many ways are there to arrange 5 friends in a line? Answer: 120 Solution: For line arrangements, it is expressed as N! = N! = 5! =5x4x3x2x1

26) Determine the number of five-digit numbers that start with the digit 7, have no digit 0, and have no repeating digits Answer: 1680 Solution: This is a conditional line arrangement. There are 9 digits from 1 to 9 (0 is not included) thus our factorial will be within these ranges. Since the first digit is 7, there is only one possibility for the first. Following the regular factorial pattern after, the second should be (9 - 1) = 8 since one digit was taken by the first. The third must be (8 - 1) = 7, the fourth must be (7 - 1) = 6, the fifth must be (6 - 1) = 5 Multiplying, =1x8x7x6x5 = 1680 27) How many three digit-numbers exist such that the product of the three digits is 18? Answer: 15 Solution: The three digit groups that form 18 when multiplied, are (1, 9, 2), (3, 3, 2), and (6, 3, 1). Note that (18,1 ,1) is not considered since 18 has 2 digits There are 6 combinations for (1,9,2) There are 3 combinations for (3,3,2) There are 6 combinations for (6, 3, 1) In total, there are 15 number combinations 28) Numbers are drawn from 1 to 40. At least how many number(s) is/are needed to be drawn to ensure that there are two numbers drawn that form a product of 24? Answer: 37 Solution: To form 24, the two numbers must be one of: 1 x 24, 2 x 12, 3 x 8, 4 x 6 There are 8 numbers in all, the rest can be drawn first thus we can draw (40 - 8) = 32 numbers without having a pair, taking out one number from each pair, still no two numbers will form 24. Now there are (32 + 4) = 36 numbers drawn. Since we considered the greatest number of draws without a product of 24, the next number drawn should make a product. (36 + 1) = 37 numbers 29) A flight of stairs has 5 steps. Kris can go up for 1 or 2 steps each time.​.​How many way(s) is / are there for Kris to go up the stairs? Answer : 8 ways Solution : There is 1 way to go up the 1st step of the stairs.(from zeroth step) There are 2 ways to go up the 2nd step of the stairs(from zeroth (1) or first (1)). There are 3 ways to go up the 3rd step of the stairs (from first (1) or second (2)). There are 5 ways to go up the 4th step of the stairs (from the second (2) or third (3)). There are 8 ways to go up the 5th step of the stairs, (from the third (3) or the fourth (5))

30) Marie wants to go from point A to point B, if he is only allowed to go up or to the right, how many way(s) is/are there? B

A Answer: 22 Solution: By counting through the passable ways of the nodal network: