Basic question type
know the total number of chickens and rabbits and the total number of legs. I'll ask how many chickens and rabbits each. The key to solving the problem in
: Use the assumption method, assuming that it is all an animal (such as all chickens or all rabbits), and then, based on the difference in the legs, you can infer the number of heads of an animal.
Problem Solving Rules:
Method 1,
Assume all are chickens, the number of rabbits = (total number of legs - total number of × 2) ÷ (the number of feet per rabbit - the number of feet per chicken);
Method 2,
Assume all are rabbits, the number of chickens = (total number of × 4 - total number of legs) ÷ (the number of feet per rabbit - the number of feet per chicken)
Example 1: There are 20 chickens and rabbits, 44 feet, how many chickens and rabbits are there?
Solution: Method 1. Assume that it is all chicken
(44 — 20 × 2) ÷ (4 — 2 )=2 (only). . . . . . The number of rabbits is
(Total number of legs - Total number of only × 2) ÷ (The number of feet per rabbit - The number of feet per chicken)
20-2=18 (Only). . . . . . The number of chickens
Method 2. Assume that all rabbits
(20 × 4-44) ÷ (4- 2) = 18 (only). . . . . . The number of chickens is
(total number × 4 - total number of legs) ÷ (number of feet per rabbit - number of feet per chicken)
Example 2. When children go boating, the big boat can take 10 people and the small boats take 6 people. The children rented a total of 15 boats. It is known that the people who take the big boats are 22 more than those who take the small boats. How many big boats are there?
Solution: Method 1. Assume that they are all small boats
Big ships: (6×15+22)÷ (6+10) = 7 (pieces); boats: 15-7=8 (pieces)
Method 2. Assume that they are all large boats
Small boats: (10×15-22)÷ (6+10) = 8 (pieces) Big ships: 15-8=7 (pieces) 20-18=2 (pieces). . . . . . The number of rabbits
Common questions
1. The difference between the total number of heads and the number of chickens and rabbits is known. Find the number of chickens and rabbits.
(1) The difference between the total number of heads and chickens and rabbits is known. When the total number of chickens is more than the total number of rabbits,
Method 1:
(number of each chicken feet × total number of heads - difference between chickens and rabbits' feet) ÷ (number of feet per chicken + number of feet per rabbit) = rabbit number;
Total heads - rabbit number = chicken number
Method 2:
(number of each rabbit feet × total number of heads + difference between chickens and rabbits' feet) ÷ (number of feet per chicken + number of feet per free) = chicken number;
Total heads - rabbit number = rabbit number
Method 2:
(number of each rabbit feet × total number of heads + difference between chickens and rabbits' feet) ÷ (number of feet per chicken + number of feet per free) = chicken number;
Total heads - chicken number = rabbit number.
Method 3:
sets the equation solution based on the difference in the number of chickens and rabbits, find out the relationship between chickens and rabbits
Example 1. There are 30 chickens and rabbits, and there are 60 more rabbits than chicken feet. How many chickens and rabbits are there?
Solution 1: Number of rabbits: (2×30+60) ÷ (2+4) = 20 (piece); Number of chickens: 30-20=10 (pieces)
Solution 2: Number of chickens: (4×30+60) ÷ (2+4) = 10 (pieces) Number of rabbits: 30-10=20 (pieces)
Solution 3: According to "the rabbit feet are 60 more than the chicken feet", that is, "the chicken feet". 60 fewer than rabbit feet”, then the number of chickens
is 2 times less than rabbits (60÷2 =) 30 (pieces)
Solution: Assuming that rabbits have X, then chickens have 2X-60÷2 (pieces) that is: 2X-30 (pieces)
2X-60÷2 + X=30
3X-30 = 30
3X=60
X=20 30-20=10 (only)
(2) The difference between the total number and the number of chickens and rabbits’ feet is known. When the total number of rabbits’ feet is more than the total number of chickens’ feet.
(number of feet per chicken × the difference between total head count + the difference between chicken and rabbit feet) ÷ (number of feet per chicken + the number of feet per rabbit) = number of rabbits; total head count - number of rabbits = number of chickens.
or (number of feet per rabbit × total number of heads - difference between chicken and rabbit feet) ÷ (number of feet per chicken + number of feet per rabbit) = number of chickens;
2. Problem of chicken and rabbit interchange (known total number of feet and total number of feet after chicken and rabbit interchange, find out how many chickens and rabbits each),
[(sum of total number of feet per chicken and rabbit feet per chicken and rabbit) + (difference between total number of feet per chicken and rabbit) ÷ (difference between total number of feet per chicken and rabbit) ÷2 = number of chickens;
[(sum of total number of feet per chicken and rabbit) ÷ (sum of total number of feet per chicken and rabbit) - (difference between total number of feet per chicken and rabbit) ÷2 = number of rabbits.
3. The solution to the gains and losses problem (promotion questions for chicken and rabbit problems) can be used as the following formula:
(1 qualified product score × total number of products - total score of actual results) ÷ (score score of each qualified product + deduct points for each unqualified product) = number of unqualified products.
or the total number of products - (deduct points from each unqualified product × total number of products + total score of actual results) ÷ (deduct points from each unqualified product +
deduct points from each unqualified product) = number of unqualified products.
Example
Example 3. There are some chickens and rabbits, and there are 44 feet in total. If the number of chickens and rabbits is exchanged, there are 52 feet in total. How many chickens and rabbits are there?
Solution: Number of chickens: [(52+44)÷(4+2)+(52-44)÷(4-2)]÷2 =20÷2=10 (pieces)
Number of rabbits: [(52+44)÷(4+2)-(52-44)÷(4-2)]÷2 =12÷2=6 (pieces)
Analysis: First, add the numbers exchanged by chickens and rabbits. Think about it, what is the result? Is the numbers of chickens and rabbits all become the total number of chickens and rabbits, and are already a small monster with six legs that has become the total number of chickens and rabbits. Therefore, (52+44)÷(4+2), the sum of chickens and rabbits is actually an ordinary problem of chickens and rabbits in the same cage. But if we look at what number is obtained by subtracting the numbers exchanged by chickens and rabbits, why is there any difference in exchange? Because rabbits have 4 legs and chickens have 2 legs, each chicken will have two extra legs if replaced with one rabbit. Therefore (52-44)÷ (4-2), the difference between chicken and rabbit is obtained. Then this becomes a problem of difference, which can be easily answered below.
Example 4. When children go rowing, the big boat can sit 10 people, the small boat can sit 6 people, and 130 people can sit 1. If the number of big boats and small boats is exchanged, the number of 20 people will be less. Ask how many big boats and how many small boats are?
Solution: Small boat: [(130-20+130)÷(10+6)+20÷(10-6)]÷2=20÷2=10 (pieces)
Large boat: [(130-20+130)÷(10+6)-20÷(10-6)]÷2=10÷2=5 (pieces)
Example 5. There are 30 chickens and rabbits in total, and there are 30 more chicken feet than rabbit feet. How many chickens and rabbits are there?
Solution: Number of rabbits: (2×30-30)÷(2+4) = 5 (piece);
Number of chickens: 30-5=25 (piece)
Analysis: First of all, assuming that they are chickens, then there are 60 feet, and then subtract the difference between the number of chickens and rabbits, then the remaining chickens and rabbits with the same number as rabbits are the same as the number of rabbits, which is also a kind of six-legged monster, so dividing by 6 will naturally result in the number of rabbits.
Example 6. When children go boating, they can take 10 people in large boats and 6 people in small boats. The children rented a total of 15 boats. It is known that 42 more people take small boats than people take large boats. How many big boats are there?
Solution: Big ship: (6×15-42)÷(6+10) = 3 (piece);
Boat: 15-3=12 (piece)
or
Boat: (10×15+42)÷(6+10) = 12 (piece)
Big ship: 15-12=3 (piece)
Total number of heads-number of chickens=number of rabbits.
Example 7. Workers in light bulbs in the light bulb factory pay wages based on the score. 4 points will be given for each product produced. Not only will the unqualified product be given no points, but 15 points will be deducted. A worker produced 1,000 light bulbs and scored a total of 3,525 points. How many of them were unqualified?
Solution 1 (4×1000-3525)÷(4+15)
=475÷19=25 (pieces)
Solution 2 1000-(15×1000+3525)÷(4+15)
=1000-18525÷19
=1000-975=25 (pieces) (answer)
(a problem of gains and losses is also called the problem of transporting glassware. If the shipment is intact, the freight is given to each one of them. If the damaged one is not given to the freight, it also needs to pay the cost of ×× yuan... Its solution can obviously apply the above formula.)
Class exercises
1. Xiaomei counts her chickens and rabbits, and there are 16 heads and 44 feet. Q: How many chickens and rabbits are there in Xiaomei’s family?
Solution: There are rabbits (44-2×16) ÷ (4-2) = 6 (pieces),
has chickens 16-6 = 10 (pieces).
Answer: There are 6 rabbits and 10 chickens.
2. 100 monks have 140 buns, one big monk has 3 buns, and one little monk has 1 bun. Question: How many people are there between the big and the little monks?
Assuming that 100 people are all monks, then a total of 300 buns are needed, which is 300-140=160 (pieces) more than the actual situation. Now, the number of people who change to the monk will remain unchanged for each change, and the steamed bun will be reduced by 3-1=2 (pieces). Because 160÷2=80, there are 80 monks and 100-80=20 (pieces).
3. Each set of colored cultural supplies costs 19 yuan, and each set of ordinary cultural supplies costs 11 yuan. A total of 16 sets of these two cultural supplies were bought, which cost 280 yuan. Q: How many sets of two types of cultural supplies have you bought?
Assuming that you buy 16 sets of colored cultural supplies, you will need 19×16=304 (yuan), which is 304-280=24 (yuan) more than the actual situation. Now you use ordinary cultural supplies to exchange for colored cultural supplies. Each set is used to exchange for 19-11=8 (yuan) less, so buy ordinary cultural supplies 24÷8=3 (sets), and
buy colored cultural supplies 16-3=13 (sets).
4. There are 100 chickens and rabbits in total, and 20 more chicken feet than rabbit feet. Q: How many chickens and rabbits are there?
Analysis: Assuming that 100 are chickens and there are no rabbits, then there are 200 chicken feet, and the number of rabbits is zero. In this way, there are 200 more chicken feet than rabbit feet, but in fact only 20 more, which shows that the assumed number of chicken feet is 200-20=180 more than rabbit feet (only). Now, we exchange rabbits for chickens. For each change of one, the chicken feet will be reduced by 2 and the rabbit feet will be increased by 4, that is, the number of feet with more chicken feet will be reduced by 4+2=6 (pieces), while 180÷6=30, so there are 30 rabbits and 100-30=70 (pieces). Solution: There are rabbits (2×100-20) ÷ (2+4) = 30 (pieces), and there are chickens 100-30 = 70 (pieces).
Answer: There are 70 chickens and 30 rabbits.
5. There are 50 large and small oil bottles in total, each large bottle can hold 4 kilograms of oil, and each small bottle can hold 2 kilograms of oil. The large bottle contains 20 kilograms more than the small bottle. Q: How many large and small bottles are there?
Solution: There are (4×50-20) ÷ (4+2) = 30 (pieces), and there are 50-30 = 20 (pieces) for large bottles of
.
Answer: There are 20 large bottles and 30 small bottles.
6. A batch of steel is loaded with a small truck and only 36 are loaded with a large truck. It is known that each large truck is equipped with 4 tons more than each small truck, so how many tons of steel are there in this batch?
Analysis: To calculate how many tons of steel this batch is, you need to know how many tons of each large truck or small truck can hold.
uses the assumption, assuming that only 36 small trucks are used to load this batch of steel, because each large truck is loaded with 4 tons more than each small truck, so 4×36=144 (tons) are left. According to the conditions, 45-36=9 (cars) of small trucks are required to load the 144 tons of steel. In this way, each small truck can be equipped with 144÷9=16 (tons). From this we can find out how many tons of steel there are.
Solution: 4×36÷(45-36)×45=720 (tons).
Answer: There are 720 tons of steel in this batch.
7. Lele Department Store entrusts the transport station to transport 500 vases. The two parties agreed that each freight cost 0.24 yuan, but if damage occurs, not only will the freight cost be paid for each broken one, but also 1.26 yuan will be compensated. As a result, the transport station will receive a total freight cost of 115.5 yuan. Q: How many vases did the Communist Party break during the transportation process?
Analysis: Assuming that no 500 vases were broken during the transport process, then the freight was 0.24×500=120 (yuan). In fact, I only get 115.5 yuan, and I get 120-115.5=4.5 (yuan). For every vase that is broken at the porting station, it will lose 0.24 + 1.26 = 1.5 (yuan). Therefore, a total of 4.5÷1.5=3 (only).
solution: (0.24×500-115.5)÷ (0.24+1.26) = 3 (only).
Answer: A total of 3 vases were broken.
8. Xiaole and Xiaoxi jumped rope together. Xiaoxi jumped for 2 minutes first, and then the two jumped for 3 minutes each, and a total of 780 jumps were jumped.It is known that Xiaoxi jumps 12 more times per minute than Xiaole, so how many more times does Xiaoxi jump than Xiaole?
Analysis and Solution: Using the hypothesis method, assuming that Xiaoxi's jumping speed is reduced to the same as Xiaole, then the total number of jumps for the two people is reduced by
12×(2+3)=60 (Part 2).
can be found that Xiaole jumps
(780-60) ÷ (2+3+3) = 90 (Part 2),
Xiaole jumps 90×3=270 (Part 2), so Xiaoxi jumps more than Xiaole
780-270×2=240 (Part 2).
After-school assignment
1. There are 100 students in a certain school participating in a mathematics competition with an average score of 63 points, including 60 points for boys and 70 points for girls. There are more male students than female students.
Girls: (63100-60100)(70-60)=30 (person)
Boys: 100-30=70 (person)
70-30=40 (person)
2. There are a bunch of black and white chess pieces, among which black chess pieces has twice the number of white chess pieces. If 4 black chess pieces and 3 white chess pieces are taken out from this pile of chess pieces at the same time each time. Then after taking out _______ times, there are 1 white zi left and 18 red zi left.
The number of blackspots is twice the number of whites. If two whites (half of blackspots) are taken out each time, then there are 18 blackspots in the end, and 18 whites should be left.
Now there is only one whites left. This is because in fact, taking 3 each time is more than assuming that one more each time, so a total of (9-1)(3-2)=8 (times)
3. Students buy 4 basketballs and 5 volleyballs, and a total of 185 yuan is spent on a student. A basketball is 8 yuan more expensive than a volleyball. The unit price of basketball is ________ yuan.
(185-48)(5+4)+8=25(yuan)
4. Xiaoqiang loves to collect stamps. He bought two stamps of 4 and 8 cents for 1 yuan, a total of 20 pieces. Then he bought 4-cent stamps________.
(208-100)(8-4)=15(sheet)
5. The squirrel mother picks pine nuts, picks 20 pieces a day on sunny days, and 12 pieces a day on rainy days. It picked 112 pieces a row, and averaged 14 pieces a day. In these days, it is rainy.
(1121420-112)(20-12)=6(days)
6. Some coins with 2 points and 5 points have a total of 299 points, of which the number of 2 points is 4 times the number of 5 points, and there are _________.
299(24+5)=23 (pieces)
7. Someone receives a salary of 240 yuan, and there are 50 RMB in total, including 2 yuan, 5 yuan and 10 yuan. Among them, 2 yuan is the same as 5 yuan, so 10 yuan has ________.
(1050-240)[10-(2+5)2]=40(sheets)
[ 240-(2+5)(402)]10=10(sheets)
8. Buy some 40 stamps with 4 cents and 8 cents in total. It is known that 8 cents stamps have 40 more stamps than 4 cents, so 8 cents stamps have ______ sheets.
4 points: (680-840)(8+4)=30(sheets)
8 points: 30+40=70(sheets)
9. There are 200 chickens and rabbits, and the feet of chickens are 56 less than those of rabbits. How many chickens are there?
rabbit: (200+562)(2+1)=76(pieces)
Chicken:200-76=124(pieces)
10. There is a truck transporting 2,000 glass bottles. The freight is calculated based on the number of intact bottles when it arrives. Each one is 2 cents. If it is damaged, it will be compensated for 1 yuan if it is damaged. As a result, the freight is 379.6 yuan. How many glasses were damaged during this transport?
(0.22000-379.6)(1+0.2)=17(only)
11. A certain math test has 20 questions. If you do one question correctly, you will get 5 points. If you do one question wrong, you will get 1 point in reverse. If you do it, you will get 0 points. Xiaohua scored 76 points. How many questions did he do correctly?
Analysis: 76 points are 24 points less than the full score, 6 points less if you do wrong one question, 5 points less if you don’t do it, 24 points can only do it wrong 4 questions, then you don’t do it, and 16 questions are correct.
12. A and B shoot, if they hit, A scores 4 points and B scores 5 points; if they fail, A loses 2 points and B scores 3 points, each of them shoots 10 shots, and a total of 14 shots are hit. When the score is settled, A scores 10 points more than B. How many shots are A and B hit?
analysis: Assume that A wins 10 shots, B wins 14-10=4 (send). A gets 410=40 (points), B gets 54-36=2 (points). The difference between the conditions for this question "A is 10 points more than B" (40-2)-10=28 (points), A scores 1 less, 4+2=6 (points), B can increase 5+3=8 (points). 28(8+6)=2 10-2=8(日)„„A. 14-8=6(日)„„B.
