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Solve Exercises Lesson 22 Proportional Quantities (Chapter 6 Math 7 Connect)
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Solve lesson 6.17, page 14, Math 7 Textbook Connecting knowledge volume 2
Let x and y be two quantities that are directly proportional. Replace each “?” in the following table with the appropriate number.
x |
2 |
4 |
5 |
? |
? |
? |
y |
-6 |
? |
? |
9 |
18 |
1.5 |
Write a formula that describes the dependence relationship between two quantities x and y.
Solution method
Using the property of two proportional quantities: \(\dfrac{{{x_1}}}{{{y_1}}} = \dfrac{{{x_2}}}{{y_2}}} = \dfrac {{{x_3}}}{{{y_3}}} = …\)
Detailed explanation
x |
2 |
4 |
5 |
-3 |
-6 |
-0.5 |
y |
-6 |
-twelfth |
-15 |
9 |
18 |
1.5 |
Since x and y are proportional quantities, there are \(\dfrac{{{y_1}}}{{{x_1}}} = \dfrac{{ – 6}}{2} = – 3\) so we has the formula y = -3. x
Solve lesson 6.18 page 14 Math 7 textbook Connecting knowledge volume 2
According to the table of values below, are the two quantities x and y proportional?
a)
x |
5 |
9 |
15 |
24 |
y |
15 |
27 |
45 |
72 |
b)
x |
4 |
8 |
16 |
25 |
y |
8 |
16 |
30 |
50 |
Solution method
Check if the ratio of their respective values is always equal.
+ If equal, then the two quantities are proportional
+ If not equal, then the two quantities are not two proportional quantities
Detailed explanation
a) We have: \(\dfrac{5}{{15}} = \dfrac{9}{{27}} = \dfrac{{15}}{{45}} = \dfrac{{24}}{ {72}}\) so 2 quantities x, y are two proportional quantities.
b) We have: \(\dfrac{4}{8} = \dfrac{8}{{16}} = \dfrac{{25}}{{50}} \ne \dfrac{{16}}{{ 30}}\) so two quantities x, y are not directly proportional.\(\dfrac{5}{{15}} = \dfrac{9}{{27}} = \dfrac{{15 }}{{45}} = \dfrac{{24}}{{72}}\)
Solve lesson 6.19 page 14 Math 7 textbook Connecting knowledge volume 2
Given that y is proportional to x by the scaling factor a, x is proportional to z by the scaling factor b. Is y proportional to z? If yes, what is the ratio?
Solution method
+ Use the definition of 2 proportional quantities:
If y = ax (a is a non-zero constant), then y is proportional to x by the scaling factor a.
+ Represent the quantity y in terms of z. If y = k. z (k is a constant) then y and z are proportional quantities.
Detailed explanation
Since y is proportional to x by the scaling factor a, y = ax
Since x is proportional to z by the scaling factor b, x = bz
Therefore, y = ax = a.(bz ) = (ab).z ( a,b are constants because a,b are constants)
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So y is proportional to z and the scaling factor is ab
Solve lesson 6.20 page 14 Math 7 textbook Connecting knowledge volume 2
Two rectangular water tanks have the same length and width respectively, but the height of the first is equal to \(\dfrac{3}{4}\) the height of the second. It takes 4.5 hours to fill the first tank with water. How long does it take to fill the second tank with water (if the same capacity pump is used)?
Solution method
The height of the water tank and the time it takes to fill the tank are two proportional quantities
Apply the property of two proportional quantities: \(\dfrac{{{x_1}}}{{{x_2}}} = \dfrac{{{y_1}}}{{y_2}}}\)
Detailed explanation
Let the time to fill the second tank with water x (hours) (x > 0)
Since two rectangular water tanks have the same length and width respectively and the pump has the same capacity, the height of the water tank and the time it takes to fill the tank are proportional.
Applying the property of two proportional quantities, we have:
\(\dfrac{3}{4} = \dfrac{{4,5}}{x} \Rightarrow x = \dfrac{{4,4,5}}{3} = 6\)( satisfied)
So the time to fill the second tank with water is 6 hours
Solve lesson 6.21 page 14 Math 7 textbook Connecting knowledge volume 2
To prepare students for the experiment, Ms. Huong divided 1.5 liters of chemicals into three parts proportional to 4, 5, 6 and stored them in three jars. How many liters of chemicals does each bottle hold?
Solution method
Let the volume of 3 parts be x,y,z (liters) respectively (x,y,z > 0)
Use the property of the series of equal ratios: \(\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = \dfrac{{a + c + e} }{{b + d + f}}\)
Detailed explanation
Let the volume of 3 parts be x,y,z (liters) respectively (x,y,z > 0)
Because Ms. Huong divided 1.5 liters of chemicals into three parts, x+y+z=1.5
Since three parts are proportional to 4;5;6 so \(\dfrac{x}{4} = \dfrac{y}{5} = \dfrac{z}{6}\)
Applying the property of the series of equal ratios, we have:
\(\begin{array}{l}\dfrac{x}{4} = \dfrac{y}{5} = \dfrac{z}{6} = \dfrac{{x + y + z}}{{ 4 + 5 + 6}} = \dfrac{{1.5}}{{15}} = 0.1\\ \Rightarrow x = 0.1.4 = 0.4\\y = 0.15 = 0.5 \\z = 0,1.6 = 0.6\end{array}\)
So 3 bottles contain 0.4 liters, 0.5 liters, and 0.6 liters of chemicals, respectively.