A general term in technical drawing refers to the ratio of the linear size of the figure in the drawing to the corresponding elements of the object. ① The formula indicating that two ratios are equal is called proportion, such as 3: 4 = 9: 12, 7: 9 = 2 1: 27. In 3: 4 = 9: 12, 3 and 12 are called external proportional terms, and 4 and 9 are called internal proportional terms. None of the four digits of the ratio can be 0. The number of proportion is the item of this proportion, and there are four items of proportion, namely two internal items and two external items; In 7: 9 = 2 1: 27, 7 and 27 are called external proportional terms, and 9 and 2 1 are called internal proportional terms. There are four items in the proportion, namely two internal items and two external items. 2 ratio? For example, both teachers and students have met the requirements. (3) Proportion, for example, among the goods sold, domestic products account for a relatively large proportion. (4) After the proportion is written in fractional form, the denominator on the left and the numerator on the right are the numerator on the left, and the denominator on the right is the external term. ⑤ In a proportion, the product of two external terms is equal to the product of two internal terms, which is called the basic property of proportion. ⑥ Similarities and differences between proportion and inverse proportion.
The relationship between the same point and different points
In direct proportion, two related quantities, one of which changes and the other changes with it. If the ratio of two corresponding numbers in two quantities is certain, then these two quantities are called direct ratio, and the relationship between them is called direct ratio. If the letters X and Y are used to represent two related quantities and K is used to represent their ratio, the proportional relationship can be expressed by the following formula: y÷x=k (certain) is inversely proportional to two related quantities, and one quantity changes, and the other quantity also changes. If the product of two corresponding numbers is constant, these two quantities are called inverse proportional quantities, and the relationship between them is called inverse proportional relationship. If the letters X and Y are used to represent two related quantities, and K is used to represent the inverse relationship of their products, it can be expressed by the following formula: x×y=k (certain) ratio is the proportion of the number of parts in a group to the total number, which is used to reflect the composition or structure of the group. Two related quantities, one of which changes and the other changes with it. In the chapter of learning ratio and proportion, whether the relationship between the two quantities can be correctly judged is the key point of proportion. In the process of solving such problems, we should firmly grasp the significance of positive and negative proportions. First, we should see that there are not two related quantities. Second, the quotient of these two quantities must have some product. The quotient is fixed, and the two quantities are directly proportional: the product is fixed, and the two quantities are inversely proportional. Secondly, when solving practical application problems, we should pay attention to the ratio and proportion, and their relationship with the score. Then synthesize what you have learned and just answer it.
Solution ratio
Proportion is divided into scale and proportion. Two expressions with equal ratios are called proportions. Judging whether two ratios can form a proportion depends on whether their ratios are equal. The four numbers that make up a proportion are called proportional terms. The two terms at both ends are called external terms of proportion, and the two terms in the middle are called internal terms of proportion. In proportion, the product of two external terms is equal to the product of two internal terms. The unknown term of finding proportion is called solution ratio. The solution ratio is solved by using the basic property of proportion, because the product of two external terms is equal to the product of two internal terms, so we can multiply the two external terms and the internal terms to solve this equation. For example: x: 3 = 9: 27 Solution: x: 3 = 9: 27 Solution: 27x = 3× 9 27x = 27x =16 Here are two math problems, try to do it! 125%: 7 = 4: x solution:125% x = 4× 71.25x = 28x = 28 ÷1.25x = 22.513. The ratio =9 ⑦ has the following properties: if a:b=c:d(b.d≠0), then there is1) ad = BC 2) b: a = d: c (a.c ≠ 0) 3) a: c = b: d; c:a = d:B4)(a+b):b =(c+d):d 5)a:(a+b)= c:(c+d)(a+b≠0，c+d≠0)6)(a-c = dk 1)∴ad=bk*d=kbd； Bc=b*dk=kbd ∴ad=bc 2) Obviously B: A = D: C =1/K 3) A: C = BK: DK = B: D; Binding attribute 2 is c: a = d: B4) ∫ a: b = c: d ∴ (a/b)+1= (c/d)+1∴ (a+b)/b = (c+d). That is, (a+b): b = (c+d): d+b ≠ 0, c+d ≠ 0, and the binding property 2 is b:(a+b)=d:(c+d) and b/(a+b) = d/(c+d). (k+1) ∴ a/(a+b) = c/(c+d) = k/k+1... ② that is, a: (a+b) = c: (c+d) a+b ≠. Subtract both sides of the equation to get (a-b)/(a+b) = (c-d)/(c+d) = (k-1)/(k+1) 7) Do this problem: a rectangle with a ratio of 2: 3 and an area of 36 square meters. Please sit in the back row if you are interested. Suppose a rectangle is 2 in width and 3 in length, then: width: 2x2=4 in length: 3x3=9 A: rectangle is 9 in length and 4 in width. 36 is decomposed into prime factors, and it is found that there are multiples of 2 and 3, and the results are obtained by using them.
decimal
Fractional division is the inverse of fractional multiplication. Calculation rule of fractional division: A divided by B (except 0) equals the reciprocal of A multiplied by B. When the divisor is less than 1, the quotient is greater than the dividend; When the divisor equals 1, the quotient equals the dividend; When the divisor is greater than 1, the quotient is less than the dividend. The law of fractional division is that the number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.
Fraction concept
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reciprocal
Countdown score
Find the reciprocal of the fraction, such as 3/4. Switch 3/4 numerator and denominator, so that the original numerator and denominator are the same. It's four-thirds. 3/4 is the reciprocal of 4/3. You can also say that 4/3 is the reciprocal of 3/4.
Reciprocal integer
Find the reciprocal of an integer, such as 12, divide 12 into several components, that is, 12/ 1, and then exchange the numerator and denominator of the fraction of 12/ 1. The original numerator is the denominator. Then11212 is the reciprocal of112. You can also say that112 is the reciprocal of12.
Decimal reciprocal
Find the reciprocal of a decimal, such as 0.25, divide 0.25 into several components, that is 1/4, and then exchange the numerator and denominator of this fraction, with the original numerator as the denominator and the original denominator as the numerator. That is, 4/ 1 and then 4/ 1 are converted into integers, that is, 4 0.25 is the reciprocal of 4. You can also say that 4 is the reciprocal of 0.25.
Equivalence problem
The reciprocal problem of 1 and 0
The reciprocal of 1 is still 1, because the numerator and denominator of the fraction11are interchanged, the original numerator is the denominator, and the original denominator is the numerator. 1/1 1 is an integer or1,so the reciprocal of1or1has no reciprocal, because the numerator and denominator of the fraction of 0/ 10 are interchanged. Yes 1/0 Because 0 is meaningless, neither divisor nor denominator can be 0, so 0 has no reciprocal.
The product of three or more numbers is 1.
More than three points (including three points), the product is 1. For example:1/2×1/3× 6 =1/2×1/3×1/9× 54 =1/2.
Reduction in the process of finding reciprocal.
In the process of finding the reciprocal, of course, a point is given, such as 14/35, and then it becomes 2/5. Finally, we find the reciprocal of 14/35 according to the method of finding the reciprocal.
Fan statistics chart
The fan-shaped statistical chart uses a whole circle to represent the total number, and the size of each fan in the circle indicates the percentage of the number of parts in the total number. The relationship between the number of each part and the total number can be clearly expressed through the fan-shaped statistical chart. Use the area of the whole circle to represent the total (unit 1), and use the sector area of the circle to represent the percentage of each part in the total.
function
Function: It can clearly reflect the relationship and proportion between the quantity of each part and the total quantity. The relationship between the sector area and its corresponding central angle is that the larger the sector area, the greater the degree of the central angle. The smaller the sector area, the smaller the degree of the central angle. The relationship between degree and percentage of fan-shaped central angle is: degree of central angle = percentage *360 degrees. Sector statistical charts can also be drawn as cylinders. The above is a fan formula.
circle
There are two definitions of a circle: the set of points whose distance from a plane to a fixed point is equal to a fixed length is called a circle. Second, a line segment on the plane rotates 360 around one end, leaving a track called a circle.
summary
Fold a circle in half along a straight line, completely overlapping. Fold in half in the other direction after unfolding. After folding, the point where these creases intersect is called the center of the circle, represented by the letter O, the line segment connecting the center of the circle with any point on the circle is called the radius, represented by the letter R, the line segment passing through the center of the circle with both ends on the circle is called the diameter, represented by the letter D, and the center of the circle determines the position, radius and diameter of the circle. In the same circle or equal circle, the radii are all equal and the diameters are all equal. The diameter is twice the radius, and the radius is 1/2 of the diameter. Expressed in letters: d=2r or r=d/2.
Correlation quantity of circle
Pi: The ratio of the circumference of a circle to its diameter and length is called pi, which is an infinite acyclic decimal. It is usually expressed by π=3. 14 15926535 ... In practical application, we only take its approximate value, that is, π≈3. 14 (generally π only takes 3,3.6554). An arc larger than a semicircle is called an upper arc, and an arc smaller than a semicircle is called a lower arc. A line segment connecting any two points on a circle is called a chord. The longest chord in a circle is the diameter. Central angle and central angle: the angle of the vertex on the center of the circle is called the central angle. The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle. Inner and outer center: the circle passing through the three vertices of the triangle is called the circumscribed circle of the triangle, and its center is called the outer center of the triangle. A circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, and its center is called the heart. Sector: On a circle, the figure enclosed by two radii and an arc is called a sector. The development diagram of the cone is a sector. The radius of this sector is called the generatrix of the cone. Letter representation of the correlation between circles-⊙ Radius -r or r (letter indicated by the radius of the outer ring of a circular ring) Arc-⌒ Diameter -D sector arc length/conic generatrix -L circumference -C area-S.
The positional relationship between a circle and other figures.
Position relationship between circle and point: Take point P and circle O as an example (let P be a point, then PO is the distance from the point to the center of the circle), where P is outside ⊙O, and PO > R；; P on ⊙O，po = r； P is within ⊙O, and PO < R. There are three kinds of positional relationships between straight lines and circles: there is no separated common point; There are two common points intersecting, and this straight line is called the secant of the circle; A circle and a straight line have a unique common tangent point. This straight line is called the tangent of the circle, and this unique common point is called the tangent point. Take straight line AB and circle O as examples (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle): AB is separated from ⊙O, and po > r；; AB is tangent to ⊙O, po = r；; AB intersects with ⊙O, and PO < R. There are five positional relationships between two circles: if there is nothing in common, one circle is called external separation outside the other circle, and it contains; If there is only one common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R respectively, and R≥r, and the center distance is P: outward separation P > R+R; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; It contains p.