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When you may use Pi, it is assumed to be 3.14.
Problem 1
(1) Calculation
{(1 + 2/3) × 4 + 5/6 - 7} × 8 - 9/10 =
Answer
31/10
(2) Find X
2014 / {2 + ( X - 9) × 8} + 7 = 26
Answer
22
(3) I connected rectangular paper of 3 cm in height 8 cm in width 16 pieces with X cm width of overlapping.
The area of the rectangle connected was 321 cm2.
Find X.
Answer
1.4 cm
Solution
321 / 3 = 107 cm
8 ×16 - 107 = 21 cm
21 / (16 - 1) =1.4 cm
(4) Among fractions that are larger than 7/9 and smaller than 6/7, find the fraction whose numerator is 13.
Answer
13/16
Solution
As 9 × 13/7 = 16.----, 7/9 = 13/16.----.
As 7 × 13/6 = 15.----, 6/7 = 13/ 15.----.
The fraction between 13/16.--- and 13/15.--- is 13/16.
Problem 2
Five parts in the figure below are painted by three colors of red, white, and yellow.
How many ways of painting are there in all?
Noted that any part should be painted in a different color from the next part.
Answer
30 ways
Solution
Five parts are assumed to be ① -⑤ as shown in the figure below.
When ① is painted in red and ② is painted in white, the colors which are painted in ① ~ ⑤ would be as the following tree diagrams and there are five ways.
Also when painting ① in red and ② in yellow, there are five ways and there are ten ways in total.
There are are ten ways also when painting ① in white and yellow.
Therefore, the painting ways are 10 × 3 = 30 ways in all.
Problem 3
Taro has a sister named Hanako who is 3 years younger than Taro.
At present, the age of Taro's father is 5 times as the sum of the age of Taro and Hanako.
Seven years later, the age of the father becomes 2 times of the sum of the age of Taro and Hanako.
How old is Taro now?
Answer
5 years old
Solution
When Taro's age is set to be X, Hanako's age is to be X-3 and his father's age is (X + X - 3) ×5 = X × 10 -15.
Seven years later father's age will be x ×10 -15 + 7 = X × 10 - 8, Taro is X + 7, and Hanako is X - 3 + 7 = X +4.
X + 7 + X + 4 = X × 2 + 11
(X × 2 + 11) × 2 = X × 4 + 22
X × 10 - 8 = X × 4 + 22
X × 6 = 30
Thus X = 5.
Problem 4
Quadrangular ABCD in the figure below is a parallelogram whose area is 72cm2.
AE = EF = FD = BG = GH = HC.
Find the area of the shadow part.
Answer
8cm2
Solution
The area of parallelogram FBGD is 72 × 1/3 = 24 cm2.
As BP : PF = 2:1, the area of the shadow part is 24 ×1/(1+2) = 8 cm2.
Problem 5
There are stairs in the place where ia 6m away from the pole that is set up in the ground vertically.
One step of vertical length of these stairs is 50cm, and the horizontal length is 50cm.
The top of the shadow of this pole is in the position of A in the figure.
The length of the shadow of 70cm stick which is set up in the ground vertically is 1.75m.
Find the length of the pole.
Answer
4.3 m
Solution
0.7 : 1.75 = 70 : 175 = 2 : 5
PQ = 7 × 2/5 = 2.8 m
The length of the pole is 1.5 + 2.8 = 4.3 m.
Problem 6
The salt melt up to 38g in 100g of hot water of 80 degrees Celsius.
Answer the following questions.
(1) The salt 19g was put in 30g of hot water of 80℃ and it is mixed well.
Find the weight of the salt that remains without melting.
(2) The hot water of 80℃ is continuously added while mixing well.
How many grams of hot water was added when all salt melts?
Moreover, find the concentration of the salt solution.
Noted that the concentration should be rounded off the 2nd decimal place and should be found the 1st decimal place.
Answer
(1) 7.6 g
(2) Hot water 20 g, Concentration 27.5%
Solution
(1) 38 × 19/38 = 11.4g
19 - 11.4 = 7.6g
(2) 100 ×19.38 = 50g
50 -30 = 20g
19 / (50 + 19) = 19 / 69 = 0.2753----
Thus it is 27.5%.
Problem 7
There are five points A, B, C, D, and E on the circumference as shown in the figure below.
AB = BC, and DE = EA.
∠A = 105 degrees
∠C － ∠D = 25 degrees
Find the size of ∠C.
Answer
125 degrees
Solution
According to the statement of the problem, angles are shown in the figure below.
In this figure, as ∠A = 105 degrees, P + Q = 105 degrees and ∠C － ∠D = 25, (P+S) - (Q+S) = P - Q = 25 degrees.
Thus P = (105 + 25) / 2 = 65 degrees and Q = 40 degrees.
The sum total of the interior angle of the pentagon is 540 degrees.
2S = 540 - (P+Q) × 4 = 540 - 420 = 120
Thus S = 120/2 = 60 degrees.
∠C = P +S = 65 + 60 = 125 degrees.
Problem 8
At a certain confectionery, sugar 50g and flour 110g are used for making one box of cookie.
Moreover, sugar 75g and flour 85g are used for making one box of madeleine.
In order to make several boxes of cookie and madeleine, 1300g of sugar and 1900g of flour are used.
How many pieces of box of madeleine made.
Answer
12 pieces
Solution
Number of pieces of box of cookie and madeleine are set to be X and Y respectively.
50X +75Y = 1300 ---① and 110X + 85Y = 1900 ---②
① × 11 - ② × 5
Then 400Y = 4800.
Therefore Y = 4800/400=12 pieces.
Problem 9
Water was poured into a container of Fig.1 by the rate of 40 cm3 per second.
Fig.2 is the chart which showed a relation between time and the depth of the water since it's started pouring until it becomes filled with water.
Answer the following questions.
(1) Find the length of X of Fig.1.
(2) Find the length of Y of Fig.1.
Answer
(1) 4 cm
(2) 28 cm
Solution
(1) In the Fig.3, 15 : 20 = 3 : 4.
In the Fig.4, A : B = 3 : 4.
(46 - 18) / ( 3 + 4) = 4
A = 3 × 4 = 12 seconds.
X = 40 × (18 + 12) / (20 × 15) = 4 cm
(2) B= 4 × 4 = 16 cm
40 ×16 = 640 cm3
640 / (20 × 4) = 8 cm
Thus Y = 20 + 8 = 28 cm.
Problem 10
The figure below is a solid which is made by hollowing out the cylinder whose diameter of the bottom is 2cm and height is 4cm from the cube whose length of one side is 4cm.
Each center of the bottoms of the hollowed cylinder is the same as the diagonal of the intersection of square ABCD and square EFGH respectively.
Point M is a middle point of the side CG.
This solid is cut by the plane that passes three points E, F and M.
Find the volume of the solid including face ABCD.
Pi is set to be 3.14.
Answer
38.58 cm3
Solution
The volume of the solid ABCD-EFMN is 4 × 4 × 4 × 3/4 = 48 cm3.
PS = 4 - ( 2 × 1/4) = 7/2 cm.
QR = 4 - ( 2 × 3/4) = 5/2 cm.
The volume of the below part of cylinder is 1 × 1 × 3.14 × (7/2 + 5/2) × 1/2 = 9.42 cm3 .
Therefore the volume to be found is 48 - 9.42 = 38.58 cm3.
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