E1 Find the number of the digit 1 in the numbers 1, 2, 3, . . , 100. 

Ans: 10 as units digit (01, 11…91)+10 as tens digit (10, 11…19) + 1 as hundreds digit (100) = 21

E2 Admission tickets to a variety show costs P80 for adults and P50 for kids. The ratio of kids to adults is 5∶2. If they paid P10 000, find the number of adults in the group.    

Ans: No exact answer. For the given answer to be correct, the problem should be the ratio of adults to kids is 5:2, in which case a set of 5 adults and 2 kids will pay 5(P80) + 2(P20) = P500; P10,000/500 = 20; 5x 20 = 100

E3 How large is a base angle of an isosceles triangle if its vertex angle is 52 ?    

If all angles are equal, they will be 60 each. The vertex angle is smaller by (60 -52) = 8 degrees. Share 8 equally to each of the base angles: (60 + 4) = 64.

E4 The volume of a cube is 216 cm3 . What is the total area of the faces?  [216 cm2]

Ans: S3 = 216; s = 6 ; SA = 6s2 = 6 x 36 = 216 cm2

E5 What is the largest 4-digit number that is a common multiple of 3 and 4? 

Ans: The largest 4-digit number is 9999 which is divisible by 3 but not 4. 10,000 is divisible by 4 so 9996 divisible by 4. It is also divisible by 3.

E6 A sack of rice weighs 50 kilograms. If the rice is to divided equally among 8 families, how many grams of rice would each family receive? [6250 grams]

Ans: soln. 1) Halve 50 3 times. 50/2 = 25; 25/2 = 12.5; 12.5/ 2 = 6.25kg = 6250 grams. soln 2) 50 can be split into 48 + 2 and (48 + 2)/6 = 6 + 2/8 = 6 + 1/4 = 6.25 kg This is equal to 6250 grams.

E7 What is the remainder when 4823 is divided by 11? 

Ans: Find the alternating sum from the right: (3 + 8) – (2 + 4) = 5

Or subtract known multiples of 11 repeatedly until the answer is known: 4823 – 4400 = 423; 423 – 330 = 93; 93 – 88 = 5.

E8 A four-digit number has three equal digits, and the sum of the digits is 35. Which one is the different digit? 

Ans: If the sum is 36, the four digits will be all 9; since it is only 35, one digit must be 8.

E9 Armand sold 40 cup cakes and Liza sold 50 cupcakes. The cupcakes were sold at the same price. If the total amount received was P6300, how much did Liza receive? [P3500]

Ans: A total of 40 + 50 = 90 cupcakes was sold for P6300 or P70 each. Liza sold 50 x P70 = P3500.

E10 To go to school, it took Krystel 1 hour and 40 minutes. She arrived at 7:25 am. What time did she leave her house? [6:45 am]

Ans: Wrong answer given. 1 hour and 40 minutes = (1 and 25) + 15 mins. If travel time is only 1H and 25 min, then Krystel can leave at 6 AM and arrive at 7:25AM. But travel time is 15 min more so Krystel must have left at 15 before 6 Am or 5:45AM

EX The sum 9 + 99 + 999 is equal to 9 times what number? 

Ans: Divide each term by 9 to get 1 + 11 + 111 = 123

AVERAGE

A1 At home, a jar contains a number of candies. Andrew ate of the 2/7 of number of candies. The next day, he ate  1/ 5 of the remaining number of candies. On the third day, he ate 1/ 2 of the remaining number of candies. If 8 candies remain, how many candies were there initially? 

Ans: When Andrew ate 2/7 of the candies only 5/7 remained. After eating 1/5 of the remaining 5/7 which is 1/7, only 4/7 will remain.  One half of that is 2/7 which is equivalent to 8. So 1/7 is 4. The total number of candies is then 4 x 7 or 28.

A2 The sum of two numbers is 78 and their difference is 14. Find the two numbers. [46, 32]

Ans: Twice the bigger number is the sum of 78 and 14 = 92; 46 is the bigger number.

Twice the smaller number is the difference of 78 – 14 = 64; 32 is the smaller number.

A3 Three-fifths of the students in class like to play football while the rest like basketball. If there are 18 students who like basketball, how many more students like football than basketball? 

Ans: 3/5 like to play football and  the rest, 2/5  or 18 students like basketball. The difference between the two groups is (3/5 – 2/5) is 1/5 or half of 2/5. The difference then is half of 18 or 9.

A4 Elsa is running at a rate of 360 meters/min. Find her rate in cm/sec.

Ans: 360 meters/ min = 360 m/60 sec = 6m/sec = 600cm/sec

A5 A basketball team scores an average of 53 points per game in its first 4 games and an average of 52 points per game in its first 5 games. How many points did the team score in its 5th game? 

Ans: An average of 53 points in the first four games is like scoring 53 in each of the four games. Likewise an average of 52 points in the 5 games is equivalent to scoring 52 in each of the games. For the 5th game, the team must score 52 less one point for each of the first four games to bring down their scores from 53 to 52. Thus the team must score 50 – 4(1) = 48.

AX Yvonne needs 3 1/3 cups of raisins for baking fruitcake. A package contains 2/ 3 of a cup of raisins. How many packages should Yvonne buy? 

Ans: 3 1/3 cups is equivalent to 10/3 cups. If a package contains 2/3 cups, Yvonne will need 10/2 = 5 cups.

Difficult

D1 The area of the shaded region is 48 cm2 . Find the area of the circular ring in terms of  π. [96π cm2]

Let R and r be the radii of the big and small circles respectively. The shaded area is equal to the difference of the areas of the big and small triangles, R2/2 –  r2/2 = (R2 –  r2)/2 =  48 cm2; (R2 –  r2) = 96 cm2. The area of the circular ring is the difference in the areas of big and the small circle: πR2 – πr2 = (R2 –  r2) π = 96π cm2

D2 Find the sum of the numbers in the 20th triple: (1, 4, 7), (2, 6, 9), (3, 8, 11), (4, 10, 13), . . .. 

Ans: The sum of the triples form an arithmetic sequence with a common difference of 5: 12, 17, 22, 27… The sum of the numbers in the 20th triple is 12 + 19(5) = 12 + 95 = 107

D3 Three guavas and two apples cost P155, while five guavas and four apples cost P285. What is the cost of each guava? [P25]

Ans: 3 guavas and 2 apples cost P155 so 6 guavas and 4 apples cost P310. Subtract 5 guavas and 4 apples which cost P285. The result is P25 for 1 guava.

D4 The figure shown is formed by 17 identical squares of side 3 cm. Find the perimeter of the figure. [108 cm]

Ans: The perimeter of 17 identical squares is 17 x (3 x 4) = 204 cm. Count the number of touching edges of the adjacent small squares in the figures: (16). Each edge is 3 so the total length of the touching edges is 16 x (3 x 2) cm = 96cm. 204 – 96 = 108 cm.

D5 Three cans and three pails can hold 18 liters of water. A pail can hold 1.5 liters more than a can. What is the capacity of one can? [2.25 liters]

Ans: 3 cans and 3 pails hold 18 liters; 1 can and 1 pail hold 6 liters. Deduct from this the difference in capacity of the pail and the can 6 – 1.5 = 4.5. Get half of it 4.5/2 = 2.25 liters.

DX A square piece of paper is folded into two halves to form two rectangles. If each rectangle has a perimeter of 27 cm, find the perimeter of the original square paper. [36 cm]

Ans: When the paper is folded into two halves to form rectangles, two additional sides are formed along the fold. Each of this side has the same length as the side of the square. Since the perimeter of each rectangle is  27, the total of the two perimeters is 2 x 27 = 54 which represents six sides. Each side is 54/6 = 9 cm. So the perimeter of the original square is 4 x 9 = 36 cm.

Tiebreaker

TB1 The value of √2019 is between what two consecutive integers? [44 and 45]

Ans: 2025 is the square of 45 while 2025 – 44 – 45 = 1936 is the square of 44. So √2019 is between 44 and 45.

TB2 Two pieces of circular tablecloths with areas 25π cm2 and 49π cm2 are overlapped as shown. There is an overlap of 10π cm2. Find the total area of the two pieces of tablecloths in terms of π. [64π cm2]

Ans: The total area is the sum of the areas of the two tablecloths minus the overlap: (25π + 49π) – 10 π = 64π cm2.

TB3 Find the area of the shaded region.

Ans: A = ( b x h) /2  =  [(20-8) x 12] / 2 = [12 x 12 ] / 2 = 72 cm2

DoD What is 0.177 ÷ 0.02?

Ans: Multiply both decimals by 100 before dividing. 0.177 ÷ 0.02 = 17.7 ÷ 2 = 8.85