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P1 Operation with fraction (distribution)
Find the number in X.
X/4 - X/7 = 3
Answer
28
Solution
X/4 - X/7 = 3
X x 1/4 - X x 1/7 = 3
According to the distributive principle, this formula is to be
X x (1/4 - 1/7) = 3
1/4 - 1/7 = 3/28
X x 3/28 = 3
∴ X = 3 / (3/28) = 28
P2 Magic square 3 × 3
As shown in the figure, five numbers are put in nine grids.
Other numbers are put into the remaining grids and it is arranged that products of three numbers in any vertical or horizontal row become same.
Find the number in A at this time.
Answer
22
P3 The volume when water freezes
P3 The volume when water freezes
When water freezes, suppose that the volume increases by one-eleventh.
Inversely, when ice is melted to become water, how much does the volume decrease?
Answer
One-twelfth
Solution
It becomes easy to understand that you think with concrete numerical value.
Assume the volume of the water is 11, when it freezes, the volume becomes to 12 by one-eleventh increase.
Because the volume becomes 11 water when ice of volume 12 melts, the volumes decrease by one-twelfth.
P4 Work done by two persons
To finish a certain work, it will take 12 days by Taro and 15 days by Jiro.
Taro and Jiro jointly started to finish the work but Jiro rested a number of days.
It took 8 days to finish the work.
How many days did Jiro rest ?
Answer
3 days
Solution
The volume of this work is set to 1.
The daily work load of Taro is 1/12 and Jiro is 1/15.
For two persons, total daily work load is 1/12 + 1/15 = 3/20.
Since the work load at the time of working for eight days by two persons is set to 3 / 20 × eight-day =6/5.
The difference between 1 and 6/5 comes to the work load Jiro rested.
It is (1/5)/(1/15) = 3 days that Jiro rested.
P5 100 m dash
Taro raced with Jiro 100m dash.
Jiro was 10m this side to the goal when Taro made a goal.
If Taro starts from 10m back from a start point, who will win ?
Answer
Taro
Solution
Since Jiro can run only 90m while Taro runs 100m, Taro and Jiro stand in a line at 10m before the goal.
While running remaining 10m, Taro wins by pulling apart Jiro 1m.
P6 Average speed
I went to Point B from point A at 3 km/h and returned to A from B at 6 km/h.
Find the speed of the average.
Answer
4 km/h
Solution
Average speed should be calculated by the formula of
(total of advanced distance) / (total of the taken time).
First of all, because we do not know the distance between A and B, it makes decisions for oneself.
It to be easily calculated, it is assumed 6km in the lowest common multiple of 3 and 6.
The time it took to go is at 6km ÷ 3km / h = 2 hours, the time of return is 6km ÷ 6km / h = 1 hour.
Since the sum of the round-trip distance is 12km, average speed is 12km / (1 hour +2 hour) = 4km / h.
P7 Buy apple and orange
I planed to buy 20 apples and oranges in total. one apple costs 70 yen and one orange costs 40 yen.
However, because I bought the number of apples and oranges adversely, 120 yen remained.
Find the amount of money planned?
Answer
1160 yen
Solution
As for buying the number of apples and oranges adversely and the remainder is 120 yen, it is understood that the number of apples was more than that of orange in the plan.
Whenever one apple is replaced with one orange, difference of the total price is 70 yen - 40 yen =30 yen.
Differing for 120 yen means that 120 yen / 30 yen = 4 apples was more than the number of oranges.
The sum is 20 and the difference is 4.
It is found that the number of apples is (20 + 4) / 2= 12 and the number of oranges is 20 - 12 = 8.
Therefore, the amount of money of the plan is 70 yen × 12 pieces + 40 yen × 8 pieces = 1160 yen.
P8 Repack oranges in the bag
There are ten oranges per bag in several bags.
When this was repacked to fifteen oranges per bag, four bags remained.
How many bags are there?
Answer
Twelve
Solution
Since the number of oranges does not change, the number of required bags and the number of oranges in one bag becomes an inverse ratio.
Since the ratio of the number of oranges in one bag is 10 pieces : 15 pieces = 2 : 3, the number of required bags is 3 : 2.
3 - 2 = 1 of the difference of this ratio is equivalent to four sheets.
Therefore, number of bags is 4 sheets × 3= 12 sheets.
P9 Fraction can not be reduced
How many fractions are there which cannot be reduced to its lowest numbers among 1/45, 2/45, 3/45 -- 44/45 ?
Answer
24 fractions
Solution
Since 45= 3 × 3 × 5, it turns out that the fraction of which denominator is 45 can be reduced by the multiple of 3 and 5.
Namely, we should find the number which is not the multiple of 3 nor 5 among the numerator from 1 to 44.
Number of the multiple of 3 is 14 by the calculation of 44/3 = 14 remainder 2.
Number of the multiple of 5 is 8 by 44/5 = 8 remainder 4.
Number of the multiple of 15 of the least common multiple of 3 and 5 is 2 by 44/15 = 2 remainder 14.
Therefore, the number of the fractions which cannot be reduced is 44 - (14 + 8 - 2) =24.
P10 Divided by 4 and 5
Among the numbers which will remain 2 when it is divided by 4, and will remain 3 when it is divided by 5, what is the largest number of two-digit numbers?
Answer
98
Solution
A number which will remain 2 when it is divided by 4 and remain 3 when it is divided by 5 is to be a number which can be divided by both of 4 and 5 when 2 is added to the number.
Such a number turns into the number which subtracted 2 from the multiple of 20 of the least common multiple of 4 and 5.
The largest number of two-digit is 20 x 5 - 2 = 98.
P11 Three digits number out of four cards
When creating the number of three digits by choosing and putting three sheets in order out of four cards, 1, 2, 4, and 7, how many kinds of multiples of 2 are made?
Answer
12 kinds
Solution
Since it comes to be a multiple of 2, ones digit is set to 2 or 4.
Thus, since the number of ones digit is two kinds, hundreds digit is three kinds except one sheet used at ones digit and tens digit can consider two kinds except two sheets used at both of ones digit and hundreds digit, and it is in all 2 × 3 × 2 = 12 kinds.
P12 Four children between parents
How many ways of forming line that four children stand in one line of side among parents are there?
Parents stand at the both ends of the line.
Answer
48 ways
Solution
The way of forming line of four children is 4 × 3 × 2 × 1 = 24 ways.
The way of forming both ends of parents is two ways of father-mother, mother-father.
Therefore, 24 × 2 = 48 ways in all.
P13 Time going up by elevator
There is a building of 25 stories and it took 5 seconds to go up from the first(base) floor to the fifth floor in an elevator.
How many seconds does going up by the same speed from the first floor to the 25th floor take?
Answer
30 seconds
Solution
5 seconds are taken in four between stories from the first floor to the fifth floor.
Since it is 24 between stories from the first floor to the 25th floor, this time is 5(sec) ÷ 4 × 24 = 30seconds.
P14 Trees planted along the road
Trees are planted from end to end along a road.
When planting every 6 m, ten trees are needed more than the case where it plants every 8 m.
Find the length of this road.
Answer
240 m
Solution
In the case when it is planted every 8m and it is planted every 6m, the ratio of the length of the interval of trees is 6m : 8m = 3 : 4.
Since the length of the road is the same in both cases, the number of interval of trees and the length of the interval of trees is inverse ratio, the ratio of the number of intervals of trees is 4 : 3.
The number of the intervals of trees of the case of every 6 m is set to 10 × 4 = 40 and the length of the road is 6 m × 40 = 240 m.
P15 Area of trapezoid
Find the area of trapezoid of the figure.
Answer
80 cm2
Solution
Trapezoid area = (central line) × (height) =10 × (4 + 4) = 80 (cm2)
P16 Overlapped two triangles
△DBE and △ABC are overlapped as shown in the figure below.
AB : DB = 7 : 3 and BC : BE = 8 : 5.
The area of △DBE is 15 cm2.
Find the area of △ABC.
Answer
56 cm2
Solution
The area ratio of △ABC and △DBE is set to (7 × 8) : (3 × 5) = 56 : 15.
Since △DBE is 15 cm2, △ABC is 15 × (56/15) = 56 cm2.
P17 Development view is square
Find the volume of the solid that can be assembled by the deployment view as shown below.
P18 Cutting a cylinder
The figure below is a solid obtained by cutting the cylinder on a slant.
Find its volume.
Pi is assumed to be 3.
P19 How many liars in four ?
Four persons, A, B, C, and D, said as follows :
A said "B is a liar."
B said "C is a liar."
C said "D is a liar."
D said "A is a liar."
How many liars are there among these four persons ?
Answer
Two persons
Solution
In case A is a liar, C is also a liar and B and D are the honest.
In case A is the honest, B and D are liars and C is the honest.
In any cases, liars are two.
P20 Move half distance of before
In order to go to a certain place 10 km from here, I'm going to move forward as follows :
First time, I move 5 km.
Second time I move 2.5 km of the half of 5 km.
Third time I move 1.25 km of half of 2.5 km. --------
When I move forward like above, how many times does it it take to reach the goal ?
Answer
Can not reach
Solution
In case of the method of this advance, there certainly remain the half distance from the place where I finish walking along to the goal.
Therefore I can not reach there.
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