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A good book on problem solving with very varied word problems and strategies on how to solve problems. Includes chapters on: Sequences, Problem-solving, Money, Percents, Algebraic Thinking, Negative Numbers, Logic, Ratios, Probability, Measurements, Fractions, Division. Each chapter’s questions are broken down into four levels: easy, somewhat challenging, challenging, and very challenging.
Revision note.
What type of ratio problems could i be asked to solve.
The ratio of cabbage leaves eaten by two rabbits, Alfred and Bob, is 8 : 4 respectively. It is known that Alfred eats 12 more cabbage leaves than Bob for a particular period of time. Find the total number of cabbage leaves eaten by the rabbits and the number that each rabbit eats individually.
The difference in the number of parts is
8 - 4 = 4 parts
This means that
4 parts = 12 cabbage leaves
Dividing both by 4
1 part = 3 cabbage leaves
Find the total number of parts
8 + 4 = 12 parts
Find the total number of cabbage leaves
12 × 3 = 36
36 cabbage leaves in total
Find the number eaten by Alfred
24 cabbage leaves
Find the number eaten by Bob
12 cabbage leaves
A particular shade of pink paint is made using 3 parts red paint, to two parts white paint.
Mark already has 36 litres of red paint, but no white paint. Calculate the volume of white paint that Mark needs to purchase in order to use all of his red paint, and calculate the total amount of pink paint this will produce.
The ratio of red to white is
Mark already has 36 litres of red, so
36 litres = 3 parts
Divide both sides by 3.
12 litres = 1 part
The ratio was 3 : 2 Find the volume of white paint, 2 parts
2 × 12 = 24
24 litres of white paint
In total there are 5 parts, so the total volume of paint will be
5 × 12 = 60
60 litres in total
In Jamie’s sock drawer the ratio of black socks to striped socks is 5 : 2 respectively. The ratio of striped socks to white socks in the drawer is 6 : 7 respectively.
Calculate the percentage of socks in the drawer that are black.
Write down the ratios
B : S = 5 : 2 S : W = 6 : 7
S features in both ratios, so we can use it as a link Multiply the B : S ratio by 3 to find an equivalent ratio Both ratios are now comparing to 6 striped socks
B : S = 15 : 6 S : W = 6 : 7
Link them together
B : S : W = 15 : 6 : 7
15 + 6 + 7 = 28
This means 15 out of 28 socks are black Find 15 out of 28 as a decimal by completing the division
Convert to a percentage Multiply by 100 and round to 3 significant figures
53.6 % of the socks are black
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Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.
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A ratio says how much of one thing there is compared to another thing.
Ratios can be shown in different ways:
Use the ":" to separate the values: | 3 : 1 | |
Or we can use the word "to": | 3 to 1 | |
Or write it like a fraction: |
A ratio can be scaled up:
Using ratios.
The trick with ratios is to always multiply or divide the numbers by the same value .
is the same as ×2 ×2 |
So the ratio of flour to milk is 3 : 2
To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4:
3 ×4 : 2 ×4 = 12 : 8
In other words, 12 cups of flour and 8 cups of milk .
The ratio is still the same, so the pancakes should be just as yummy.
The examples so far have been "part-to-part" (comparing one part to another part).
But a ratio can also show a part compared to the whole lot .
Part-to-Part:
The ratio of boys to girls is 2:3 or 2 / 3
The ratio of girls to boys is 3:2 or 3 / 2
Part-to-Whole:
The ratio of boys to all pups is 2:5 or 2 / 5
The ratio of girls to all pups is 3:5 or 3 / 5
We can use ratios to scale drawings up or down (by multiplying or dividing).
The height to width ratio of the Indian Flag is So for every (inches, meters, whatever) of height | |
If we made the flag 20 inches high, it should be 30 inches wide. If we made the flag 40 cm high, it should be 60 cm wide (which is still in the ratio 2:3) |
This horse in real life is 1500 mm high and 2000 mm long, so the ratio of its height to length is
1500 : 2000
What is that ratio when we draw it at 1/10th normal size?
1500 : 2000 | = 1500 : 2000 | |
= |
We can make any reduction/enlargement we want that way.
Allie measured her foot and it was 21cm long, and then she measured her Mother's foot, and it was 24cm long. |
"I must have big feet, my foot is nearly as long as my Mom's!"
But then she thought to measure heights, and found she is 133cm tall, and her Mom is 152cm tall.
In a table this is:
Allie | Mom | |
Length of Foot: | 21cm | 24cm |
Height: | 133cm | 152cm |
The "foot-to-height" ratio in fraction style is:
Allie: | Mom: |
We can simplify those fractions like this:
And we get this (please check that the calcs are correct):
"Oh!" she said, "the Ratios are the same".
"So my foot is only as big as it should be for my height, and is not really too big."
You can practice your ratio skills by Making Some Chocolate Crispies
Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. The ratio and proportion are the two important concepts, and it is the foundation to understand the various concepts in mathematics as well as in science.
In our daily life, we use the concept of ratio and proportion such as in business while dealing with money or while cooking any dish, etc. Sometimes, students get confused with the concept of ratio and proportion. In this article, the students get a clear vision of these two concepts with more solved examples and problems.
For example, ⅘ is a ratio and the proportion statement is 20/25 = ⅘. If we solve this proportional statement, we get:
20 x 5 = 25 x 4
Check: Ratio and Proportion PDF
Therefore, the ratio defines the relationship between two quantities such as a:b, where b is not equal to 0. Example: The ratio of 2 to 4 is represented as 2:4 = 1:2. And the statement is said to be in proportion here. The application of proportion can be seen in direct proportion .
The definition of ratio and proportion is described here in this section. Both concepts are an important part of Mathematics. In real life also, you may find a lot of examples such as the rate of speed (distance/time) or price (rupees/meter) of a material, etc, where the concept of the ratio is highlighted.
Proportion is an equation that defines that the two given ratios are equivalent to each other. For example, the time taken by train to cover 100km per hour is equal to the time taken by it to cover the distance of 500km for 5 hours. Such as 100km/hr = 500km/5hrs.
Let us now learn Maths ratio and proportion concept one by one.
In certain situations, the comparison of two quantities by the method of division is very efficient. We can say that the comparison or simplified form of two quantities of the same kind is referred to as a ratio. This relation gives us how many times one quantity is equal to the other quantity. In simple words, the ratio is the number that can be used to express one quantity as a fraction of the other ones.
The two numbers in a ratio can only be compared when they have the same unit. We make use of ratios to compare two things. The sign used to denote a ratio is ‘:’.
A ratio can be written as a fraction, say 2/5. We happen to see various comparisons or say ratios in our daily life.
Hence, the ratio can be represented in three different forms, such as:
Key Points to Remember:
Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.
For example, the time taken by train to cover 100km per hour is equal to the time taken by it to cover the distance of 500km for 5 hours. Such as 100km/hr = 500km/5hrs.
Ratio and proportions are said to be faces of the same coin. When two ratios are equal in value, then they are said to be in proportion . In simple words, it compares two ratios. Proportions are denoted by the symbol ‘::’ or ‘=’.
The proportion can be classified into the following categories, such as:
Inverse proportion, continued proportion.
Now, let us discuss all these methods in brief:
The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two quantities, then the direction proportion is written as a∝b.
The inverse proportion describes the relationship between two quantities in which an increase in one quantity leads to a decrease in the other quantity. Similarly, if there is a decrease in one quantity, there is an increase in the other quantity. Therefore, the inverse proportion of two quantities, say “a” and “b” is represented by a∝(1/b).
Consider two ratios to be a: b and c: d .
Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.
For the given ratio, the LCM of b & c will be bc.
Thus, multiplying the first ratio by c and the second ratio by b, we have
First ratio- ca:bc
Second ratio- bc: bd
Thus, the continued proportion can be written in the form of ca: bc: bd
Now, let us learn the Maths ratio and proportion formulas here.
Assume that, we have two quantities (or two numbers or two entities) and we have to find the ratio of these two, then the formula for ratio is defined as;
where a and b could be any two quantities .
Here, “a” is called the first term or antecedent , and “b” is called the second term or consequent .
Example: In ratio 4:9, is represented by 4/9, where 4 is antecedent and 9 is consequent.
If we multiply and divide each term of ratio by the same number (non-zero), it doesn’t affect the ratio.
Example: 4:9 = 8:18 = 12:27
Also, read: Ratio Formula
Now, let us assume that, in proportion, the two ratios are a:b & c:d. The two terms ‘b’ and ‘c’ are called ‘means or mean term,’ whereas the terms ‘a’ and ‘d’ are known as ‘ extremes or extreme terms.’
Example: Let us consider one more example of a number of students in a classroom. Our first ratio of the number of girls to boys is 3:5 and that of the other is 4:8, then the proportion can be written as:
3 : 5 :: 4 : 8 or 3/5 = 4/8
Here, 3 & 8 are the extremes, while 5 & 4 are the means.
Note: The ratio value does not affect when the same non-zero number is multiplied or divided on each term.
The following are the important properties of proportion:
To understand the concept of ratio and proportion, go through the difference between ratio and proportion given here.
S.No | Ratio | Proportion |
1 | The ratio is used to compare the size of two things with the same unit | The proportion is used to express the relation of two ratios |
2 | It is expressed using a colon (:), slash (/) | It is expressed using the double colon (::) or equal to the symbol (=) |
3 | It is an expression | It is an equation |
4 | Keyword to identify ratio in a problem is “to every” | Keyword to identify proportion in a problem is “out of” |
If a : b = c : d, then:
If (a:b)>(c:d) = (a/b>c/d)
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
If a:b is a ratio, then:
Let us learn here some rules and tricks to solve problems based on ratio and proportion topics.
Example 1:
Are the ratios 4:5 and 8:10 said to be in Proportion?
4:5= 4/5 = 0.8 and 8: 10= 8/10= 0.8
Since both the ratios are equal, they are said to be in proportion.
Are the two ratios 8:10 and 7:10 in proportion?
8:10= 8/10= 0.8 and 7:10= 7/10= 0.7
Since both the ratios are not equal, they are not in proportion.
Example 3:
Given ratio are-
Find a: b: c.
Multiplying the first ratio by 5, second by 3 and third by 6, we have
a:b = 10: 15
b:c = 15 : 6
c:d = 6 : 24
In the ratio’s above, all the mean terms are equal, thus
a:b:c:d = 10:15:6:24
Check whether the following statements are true or false.
a] 12 : 18 = 28 : 56
b] 25 people : 130 people = 15kg : 78kg
The given statement is false.
12 : 18 = 12 / 18 = 2 / 3 = 2 : 3
28 : 56 = 28 / 56 = 1 / 2 = 1 : 2
They are unequal.
b] 25 persons : 130 persons = 15kg : 78kg
The given statement is true.
25 people : 130 people = 5: 26
15kg : 78kg = 5: 26
They are equal.
The earnings of Rohan is 12000 rupees every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 rupees. Find the ratio of the savings.
Savings of Rohan per month = Rs (12000-9960) = Rs. 2040
Yearly income of Anish = Rs. 191520
Hence, the monthly income of Anish = Rs. 191520/12 = Rs. 15960.
So, the savings of Anish per month = Rs (15960 – 9960) = Rs. 6000
Thus, the ratio of savings of Rohan and Anish is Rs. 2040: Rs.6000 = 17: 50.
Twenty tons of iron is Rs. 600000 (six lakhs). What is the cost of 560 kilograms of iron?
1 ton = 1000 kg 20 tons = 20000 kg The cost of 20000 kg iron = Rs. 600000 The cost of 1 kg iron = Rs{600000}/ {20000} = Rs. 30 The cost of 560 kg iron = Rs 30 × 560 = Rs 16800
The dimensions of the rectangular field are given. The length and breadth of the rectangular field are 50 meters and 15 meters. What is the ratio of the length and breadth of the field?
Length of the rectangular field = 50 m
Breadth of the rectangular field = 15 m
Hence, the ratio of length to breadth = 50: 15
⇒ 50: 15 = 10: 3.
Thus, the ratio of length and breadth of the rectangular field is 10:3.
Obtain a ratio of 90 centimetres to 1.5 meters.
The given two quantities are not in the same units.
Convert them into the same units.
1.5 m = 1.5 × 100 = 150 cm
Hence, the required ratio is 90 cm: 150 cm
⇒ 90: 150 = 3: 5
Therefore, the ratio of 90 centimetres to 1.5 meters is 3: 5.
There exists 45 people in an office. Out of which female employees are 25 and the remaining are male employees. Find the ratio of
a] The count of females to males.
b] The count of males to females.
Count of females = 25
Total count of employees = 45
Count of males = 45 – 25 = 20
The ratio of the count of females to the count of males
The count of males to the count of females
Example 10:
Write two equivalent ratios of 6: 4.
Given Ratio : 6: 4, which is equal to 6/4.
Multiplying or dividing the same numbers on both numerator and denominator, we will get the equivalent ratio.
⇒(6×2)/(4×2) = 12/8 = 12: 8
⇒(6÷2)/(4÷2) = 3/2 = 3: 2
Therefore, the two equivalent ratios of 6: 4 are 3: 2 and 12: 8
Example 11 :
Out of the total students in a class, if the number of boys is 5 and the number of girls is 3, then find the ratio between girls and boys.
The ratio between girls and boys can be written as 3:5 (Girls: Boys). The ratio can also be written in the form of factor like 3/5.
Example 12:
Two numbers are in the ratio 2 : 3. If the sum of numbers is 60, find the numbers.
Given, 2/3 is the ratio of any two numbers.
Let the two numbers be 2x and 3x.
As per the given question, the sum of these two numbers = 60
So, 2x + 3x = 60
Hence, the two numbers are;
2x = 2 x 12 = 24
3x = 3 x 12 = 36
24 and 36 are the required numbers.
Maths ratio and proportion are used to solve many real-world problems. Register with BYJU’S – The Learning App and get solutions for many difficult questions in easy methodology and followed by the step-by-step procedure.
What is the ratio give an example., what is a proportion give example, how to solve proportions with examples, what is the concept of ratios, what are the two different types of proportions.
The two different types of proportions are: Direct Proportion Inverse Proportion
Yes, we can express ratio in terms of fractions. For example, 3: 4 can be expressed as 3/4.
The formula for ratio is: x:y ⇒ x/y, where x is the first term and y is the second term. The formula for proportion is: p: q :: r : s ⇒ p/q = r/s, Where p and r are the first term in the first and second ratio q and s are the second term and in the first and second ratio.
In the given proportion 1: 2 :: 3: 4, Means are 2 and 3 Extremes are 1 and 4.
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After completing this section, you should be able to:
Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.
Note there are three different ways to write a ratio , which is a comparison of two numbers that can be written as: /**/(\$ 1 =0.82\,{€})/**/, how many dollars should you receive? Round to the nearest cent if necessary.
Solving a proportion involving weights on different planets.
A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. How much would a typical racehorse (1,000 pounds) weigh on Mars? Round your answer to the nearest tenth.
Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights are both in the denominator.
170 64 = 1,000 x 170 64 = 1,000 x
Step 2: Cross multiply, and then divide to solve.
170 x = 1,000 ( 64 ) 170 x = 64,000 170 x 170 = 64,000 170 x = 376.5 170 x = 1,000 ( 64 ) 170 x = 64,000 170 x 170 = 64,000 170 x = 376.5
So the 1,000-pound horse would weigh about 376.5 pounds on Mars.
Example 5.32, solving a proportion involving baking.
A cookie recipe needs /**/1{\text{ inch}} =/**/ how many miles). Then use that scale to determine the approximate lengths of the other borders of the state of Wyoming.
Solving a scaling problem involving model cars.
Die-cast NASCAR model cars are said to be built on a scale of 1:24 when compared to the actual car. If a model car is 9 inches long, how long is a real NASCAR automobile? Write your answer in feet.
The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths are both in the denominator.
1 24 = 9 x 24 ( 9 ) = x 216 = x 1 24 = 9 x 24 ( 9 ) = x 216 = x
This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from inches to feet is really another proportion!). The final answer is:
216 12 = 18 216 12 = 18
The NASCAR automobile is 18 feet long.
Check your understanding.
/**/16:12/**/ | /**/12:16/**/ | /**/4:3/**/ |
/**/16:12/**/ | /**/12:16/**/ | /**/16:28/**/ | /**/28:12/**/ | None of these |
625 | 6,250 | 3,456 | 345.6 | None of these |
1 | 2 | 3 | 4 | |
128 | 460 | /**/b/**/ | 541 | |
$163.84 | /**/a/**/ | $277.76 | /**/c/**/ |
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Multiply fractions.
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Multi-objective optimization for finding main design factors of a two-stage helical gearbox with second-stage double gear sets using the eamr method.
2. optimization problem, 2.1. calculation of gearbox mass, 2.2. calculation of gearbox efficiency, 2.3. objective functions and constrains, 2.3.1. objectives functions, 2.3.2. constrains, 3. methodology, 3.1. method to solve the multi-objective optimization, 3.2. method to solve mcdm problem:, 3.3. method to find the weight of criteria:, 4. single-objective optimization, 5. multi-objective optimization, 6. conclusions, author contributions, data availability statement, acknowledgments, conflicts of interest.
Click here to enlarge figure
Parameter | Nomenclature | Units |
---|---|---|
Allowable contact stress of stages i (i = 1 ÷ 2) | AS | Mpa |
Allowable shear stress of shaft material | [τ] | MPa |
Arc of approach on i stage | - | |
Arc of recess on i stage | - | |
Base circle radius of the pinion | mm | |
Base circle radius of the gear | mm | |
Center distance of stage 1 | a | mm |
Center distance of stage 2 | a | mm |
Coefficient of wheel face width of stage 1 | X | - |
Coefficient of wheel face width of stage 2 | X | - |
Coefficient of gear material | k | Mpa |
Contacting load ratio for pitting resistance | k | - |
Diameter of shaft i | d | mm |
Efficiency of a helical gearbox | η | - |
Efficiency of the i stage of the gearbox | η | - |
Efficiency of a helical gear unit | η | - |
Efficiency of a rolling bearing pair | η | - |
Friction coefficient | f | - |
Friction coefficient of bearing | f | - |
Gearbox ratio (or total gearbox ratio) | u | |
Gear ratio of stage 1 | u | - |
Gear ratio of stage 2 | u | - |
Gear width of stage 1 | b | mm |
Gear width of stage 2 | b | mm |
Gearbox mass | m | kg |
Gear mass | m | kg |
Shaft mass | m | kg |
Gearbox housing mass | m | kg |
Gear mass of stage 1 | m | kg |
Gear mass of stage 2 | m | kg |
Hydraulic moment of power losses | T | Nm |
ISO Viscosity Grade number | VG | - |
Length of shaft i | l | mm |
Load of bearing i | Fi | N |
Mass density of gearbox housing materials | kg/m | |
Mass of shaft j (j = 1 ÷ 3) | m | kg |
Mass density of shaft material | kg/m | |
Outside radius of the pinion | mm | |
Outside radius of the gear | mm | |
Output torque | T | Nmm |
Pitch diameter of the pinion of stage 1 | d | mm |
Pitch diameter of the gear of stage 2 | d | mm |
Pitch diameter of the pinion of stage 2 | d | mm |
Pitch diameter of the gear of stage 2 | d | mm |
Power loss in the gears | Plg | Kw |
Power loss in the bearings | Plb | Kw |
Power loss in the seals | Pls | Kw |
Power loss in the idle motion | Pzo | Kw |
Pressure angle | α | rad. |
Peripheral speed of bearing | v | m/s |
Sliding velocity of gear | v | m/s |
Total power loss in the gearbox | Pl | - |
Torque on the pinion of stage i (i = 1 ÷ 2) | T | Nmm |
Volume coefficients of the pinion | e | - |
Volume coefficients of the gear | e | - |
Volume of gearbox housing | V | dm |
Volumes of bottom housing A | V | dm |
Volumes of bottom housing B | V | dm |
Volumes of bottom housing B | V | dm |
Weight density of gear materials | kg/m |
Parameter | Symbol | Lower Limit | Upper Limit |
---|---|---|---|
Gearbox ratio of first stage | u | 1 | 9 |
CWFW of stage 1 | X | 0.25 | 0.4 |
CWFW of stage 2 | X | 0.25 | 0.4 |
u | u | |
---|---|---|
Lower Limit | Upper Limit | |
10 | 1.76 | 2.4 |
15 | 2.49 | 2.99 |
20 | 3.17 | 3.52 |
25 | 3.76 | 4.01 |
30 | 4.19 | 4.63 |
35 | 4.58 | 5.23 |
40 | 4.93 | 5.80 |
Trial. | u | X | X | m (kg) | η (%) |
---|---|---|---|---|---|
1 | 4.29 | 0.25 | 0.25 | 222.74 | 96.04 |
2 | 4.29 | 0.25 | 0.29 | 226.18 | 95.97 |
3 | 4.29 | 0.25 | 0.33 | 229.60 | 95.92 |
4 | 4.29 | 0.25 | 0.36 | 232.98 | 95.86 |
5 | 4.29 | 0.25 | 0.40 | 236.32 | 95.79 |
6 | 4.29 | 0.29 | 0.25 | 223.95 | 95.45 |
… | |||||
25 | 4.29 | 0.40 | 0.40 | 240.91 | 93.04 |
26 | 4.31 | 0.25 | 0.25 | 222.69 | 96.02 |
27 | 4.31 | 0.25 | 0.29 | 226.13 | 95.97 |
… | |||||
51 | 4.34 | 0.25 | 0.25 | 222.65 | 96.00 |
52 | 4.34 | 0.25 | 0.29 | 226.08 | 95.95 |
53 | 4.34 | 0.25 | 0.33 | 229.50 | 95.88 |
… | |||||
76 | 4.36 | 0.25 | 0.25 | 222.60 | 95.98 |
77 | 4.36 | 0.25 | 0.29 | 226.04 | 95.93 |
78 | 4.36 | 0.25 | 0.33 | 229.45 | 95.86 |
… | |||||
101 | 4.38 | 0.25 | 0.25 | 222.56 | 95.95 |
102 | 4.38 | 0.25 | 0.29 | 225.99 | 95.91 |
103 | 4.38 | 0.25 | 0.33 | 229.41 | 95.84 |
… | |||||
123 | 4.38 | 0.40 | 0.33 | 234.09 | 93.01 |
124 | 4.38 | 0.40 | 0.36 | 237.45 | 92.95 |
125 | 4.38 | 0.40 | 0.40 | 240.77 | 92.88 |
Trial. | n | v | G | Ri | Rank | |||
---|---|---|---|---|---|---|---|---|
m | η | m | η | m | η | |||
1 | 0.9246 | 1.0000 | 0.4517 | 0.5114 | 0.4517 | 0.5114 | 1.1322 | 3 |
2 | 0.9389 | 0.9993 | 0.4587 | 0.5111 | 0.4587 | 0.5111 | 1.1142 | 15 |
3 | 0.9531 | 0.9988 | 0.4656 | 0.5108 | 0.4656 | 0.5108 | 1.0970 | 29 |
4 | 0.9671 | 0.9981 | 0.4725 | 0.5105 | 0.4725 | 0.5105 | 1.0804 | 48 |
5 | 0.9809 | 0.9974 | 0.4793 | 0.5101 | 0.4793 | 0.5101 | 1.0644 | 75 |
6 | 0.9296 | 0.9939 | 0.4542 | 0.5083 | 0.4542 | 0.5083 | 1.1192 | 7 |
… | ||||||||
25 | 1.0000 | 0.9688 | 0.4886 | 0.4955 | 0.4886 | 0.4955 | 1.0141 | 121 |
26 | 0.9244 | 0.9998 | 0.4516 | 0.5113 | 0.4516 | 0.5113 | 1.1322 | 1 |
27 | 0.9386 | 0.9993 | 0.4586 | 0.5111 | 0.4586 | 0.5111 | 1.1144 | 12 |
… | ||||||||
51 | 0.9242 | 0.9996 | 0.4515 | 0.5112 | 0.4515 | 0.5112 | 1.1322 | 4 |
52 | 0.9384 | 0.9991 | 0.4585 | 0.5110 | 0.4585 | 0.5110 | 1.1144 | 11 |
53 | 0.9526 | 0.9983 | 0.4654 | 0.5106 | 0.4654 | 0.5106 | 1.0970 | 27 |
… | ||||||||
76 | 0.9240 | 0.9994 | 0.4514 | 0.5111 | 0.4514 | 0.5111 | 1.1322 | 2 |
77 | 0.9383 | 0.9989 | 0.4584 | 0.5108 | 0.4584 | 0.5108 | 1.1144 | 14 |
78 | 0.9524 | 0.9981 | 0.4653 | 0.5105 | 0.4653 | 0.5105 | 1.0970 | 26 |
… | ||||||||
101 | 0.9238 | 0.9991 | 0.4514 | 0.5110 | 0.4514 | 0.5110 | 1.1320 | 5 |
102 | 0.9381 | 0.9986 | 0.4583 | 0.5107 | 0.4583 | 0.5107 | 1.1144 | 13 |
103 | 0.9523 | 0.9979 | 0.4652 | 0.5104 | 0.4652 | 0.5104 | 1.0970 | 30 |
… | ||||||||
123 | 0.9717 | 0.9685 | 0.4747 | 0.4953 | 0.4747 | 0.4953 | 1.0433 | 100 |
124 | 0.9856 | 0.9678 | 0.4816 | 0.4950 | 0.4816 | 0.4950 | 1.0279 | 115 |
125 | 0.9994 | 0.9671 | 0.4883 | 0.4946 | 0.4883 | 0.4946 | 1.0129 | 125 |
No. | u | ||||||
---|---|---|---|---|---|---|---|
10 | 15 | 20 | 25 | 30 | 35 | 40 | |
u | 2.04 | 2.74 | 3.36 | 3.85 | 4.31 | 4.76 | 5.16 |
X | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
X | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |
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Dinh, V.-T.; Tran, H.-D.; Vu, D.-B.; Vu, D.; Vu, N.-P.; Do, T.-T. Multi-Objective Optimization for Finding Main Design Factors of a Two-Stage Helical Gearbox with Second-Stage Double Gear Sets Using the EAMR Method. Symmetry 2024 , 16 , 783. https://doi.org/10.3390/sym16070783
Dinh V-T, Tran H-D, Vu D-B, Vu D, Vu N-P, Do T-T. Multi-Objective Optimization for Finding Main Design Factors of a Two-Stage Helical Gearbox with Second-Stage Double Gear Sets Using the EAMR Method. Symmetry . 2024; 16(7):783. https://doi.org/10.3390/sym16070783
Dinh, Van-Thanh, Huu-Danh Tran, Duc-Binh Vu, Duong Vu, Ngoc-Pi Vu, and Thi-Tam Do. 2024. "Multi-Objective Optimization for Finding Main Design Factors of a Two-Stage Helical Gearbox with Second-Stage Double Gear Sets Using the EAMR Method" Symmetry 16, no. 7: 783. https://doi.org/10.3390/sym16070783
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Ratio problem solving is a collection of ratio and proportion word problems that link together aspects of ratio and proportion into more real life questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem. ...
Ratio problem solving is a collection of word problems that link together aspects of ratio and proportion into more real life questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem.
Ratio Word Problems: relating different things using ratios and algebra, how to solve ratio word problems that have two-term ratios or three-term ratios, How to solve proportion word problems, questions and answers, with video lessons, examples and step-by-step solutions.
A ratio is a comparison of two quantities. A proportion is an equality of two ratios. To write a ratio: Determine whether the ratio is part to part or part to whole. Calculate the parts and the whole if needed. Plug values into the ratio. Simplify the ratio if needed.
Do you want to learn how to compare and measure different quantities using ratios and rates? Khan Academy's pre-algebra course offers you a comprehensive introduction to these concepts, with interactive exercises and videos. You will also learn how to use proportions to solve word problems and graph proportional relationships. Join Khan Academy and start your journey to master ratios and rates!
Discover how to solve ratio problems with a real-life example involving indoor and outdoor playtimes. Learn to use ratios to determine the number of indoor and outdoor playtimes in a class with a 2:3 ratio and 30 total playtimes. ... What you need to do in any word problem involving the ratios is exactly the same. Take the entire amount and ...
In the classroom, ratio problem solving often comes in the form of real world scenarios or word problems. For example, \cfrac{8}{10} students are right handed. What is the ratio of left handed students to right handed students? (2 : 8) Step-by-step guide: Ratio problem solving.
Practice Questions. Previous: Percentages of an Amount (Non Calculator) Practice Questions. Next: Rotations Practice Questions. The Corbettmaths Practice Questions on Ratio.
Practise solving ratio problems and explore the different forms they take with BBC Bitesize Maths. For students between the ages of 11 and 14.
The Corbettmaths Textbook Exercise on Ratio: Problem Solving. Welcome; Videos and Worksheets; Primary; 5-a-day. 5-a-day GCSE 9-1; 5-a-day Primary; 5-a-day Further Maths ... Ratio: Problem Solving Textbook Exercise. Click here for Questions. Textbook Exercise. Previous: Ratio: Difference Between Textbook Exercise. Next: Reflections Textbook ...
The first step is to work out how many equal parts there are. The ratio is 2:1 so there are 3 (2 + 1) equal parts. We now need to work out how much each of the parts is worth. We divide the £120 between the 3 parts. £120 ÷ 3 = £40. Each part is worth £40. Now we can work out how much Abbie and Ben get. Abbie has 2 parts so she gets £80 (2 ...
You are here: Home → Worksheets → Ratios Free worksheets for ratio word problems. Find here an unlimited supply of worksheets with simple word problems involving ratios, meant for 6th-8th grade math. In level 1, the problems ask for a specific ratio (such as, "Noah drew 9 hearts, 6 stars, and 12 circles. What is the ratio of circles to hearts?
Sharing an amount in a given ratio Further problems involving ratio include Ratios where you are given the difference between the two parts. E.g., Kerry is given $30 more than Kacey who is given $50; Ratios where one quantity is given and you have to find the other quantity. E.g., Kerry and Kacey are sharing money in the ratio 8 : 5, Kacey gets $50
Solving Ratio Problems. We add the parts of the ratio to find the total number of parts. There are 2 + 3 = 5 parts in the ratio in total. To find the value of one part we divide the total amount by the total number of parts. 50 ÷ 5 = 10. We multiply the ratio by the value of each part. 2:3 multiplied by 10 gives us 20:30.
Unit test. Level up on all the skills in this unit and collect up to 1,400 Mastery points! Ratios let us see how two values relate, especially when the values grow or shrink together. From baking recipes to sports, these concepts find their way into our lives on a daily basis.
Thinking Blocks Ratios - Learning Connections. Essential Skills. Problem Solving - model and solve word problems. Common Core Connection for Grades 6 and 7. Understand the concept of ratio and describe the relationship between two quantities. Use ratio and rate reasoning to solve real-world and mathematical problems.
Using Ratios. The trick with ratios is to always multiply or divide the numbers by the same value. Example: A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. So the ratio of flour to milk is 3 : 2. To make pancakes for a LOT of people we might need 4 times the quantity, so we multiply the numbers by 4: 3 ×4 : 2 ×4 = 12 : 8.
Examples. For example, ⅘ is a ratio and the proportion statement is 20/25 = ⅘. If we solve this proportional statement, we get: 20/25 = ⅘. 20 x 5 = 25 x 4. 100 = 100. Check: Ratio and Proportion PDF. Therefore, the ratio defines the relationship between two quantities such as a:b, where b is not equal to 0.
Use proportions to solve scaling problems. Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is ...
Learn how to solve problems involving ratio and proportion with Dr Austin Maths, a website that provides free and editable resources for maths teachers and students. You can also find topics such as equations, inequalities, graphs, calculus and statistics on the website.
The ratio 3 to 5 or 3/5 is the same thing as 12 to 20, is the same thing as 24 to 40, is the same thing as 48 to 80. Let's make sure we got the right answer. Let's do a couple more of these. The following table shows equivalent fractions to 27/75. So then they wrote all of the different equivalent fractions.
Multiplication, division, fractions, and logic games that boost sixth grade math skills. Level 6 Math Games Game Spotlight: Dirt Bike Proportions Multiplayer Math Games Four Wheel Fracas. Jumping Aliens. Martian Hoverboards. Otter Rush. ... Logic and Problem Solving Games Icy Super Slide. Arcade Golf. Rabbit Samurai 2. Duck Life 4. Icy Purple ...
Equivalent ratio word problems. A fruit basket is filled with 8 bananas, 3 oranges, 5 apples, and 6 kiwis. Complete the ratio. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education ...
When optimizing a mechanical device, the symmetry principle provides important guidance. Minimum gearbox mass and maximum gearbox efficiency are two single objectives that need to be achieved when designing a gearbox, and they are not compatible. In order to address the multi-objective optimization (MOO) problem with the above single targets involved in building a two-stage helical gearbox ...
Proportion word problems. Sam used 6 loaves of elf bread on an 8 day hiking trip. He wants to know how many loaves of elf bread ( b) he should pack for a 12 day hiking trip if he eats the same amount of bread each day. How many loaves of elf bread should Sam pack for a 12 day trip? Learn for free about math, art, computer programming, economics ...
This unit tackles the medium-difficulty problem solving and data analysis questions on the SAT Math test. Work through each skill, taking quizzes and the unit test to level up your mastery progress. ... Ratios, rates, and proportions: medium Get 3 of 4 questions to level up! Unit conversion: medium. Learn. Unit conversion | Lesson (Opens a modal)