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Solve Exercises Lesson 23 Quantities of Inverse Proportion (Chapter 6 Math 7 Connect)
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Solve lesson 6.22 on page 18 Math 7 Textbook Connecting knowledge volume 2
Let x and y be two quantities that are inversely proportional. Replace each “?” in the following table with the appropriate number.
x |
2 |
4 |
5 |
? |
? |
? |
y |
-6 |
? |
? |
3 |
ten |
0.5 |
Write a formula that describes the dependence relationship between two quantities x and y.
Solution method
Use the property of two quantities that are inversely proportional: x_{first}y_{first} = x_{2}y_{2}=….
Detailed explanation
x |
2 |
4 |
5 |
-4 |
-1.2 |
-24 |
y |
-6 |
-3 |
-2.4 |
3 |
ten |
0.5 |
Since x and y are two quantities that are inversely proportional, there is x_{first}y_{first} = 2.(-6) = -12 so we have the formula \(y = \dfrac{{ – 12}}{x}\)
Solve lesson 6.23 page 18 Math 7 textbook Connecting knowledge volume 2
According to the table of values below, are the two quantities x and y two quantities in inverse proportion?
a)
x |
3 |
6 |
16 |
24 |
y |
160 |
80 |
30 |
20 |
b)
x |
4 |
8 |
25 |
32 |
y |
160 |
80 |
26 |
20 |
Solution method
Check if the product of their respective values is always equal.
If equal, then the two quantities are inversely proportional
+ If not equal, then the two quantities are not two quantities that are inversely proportional
Detailed explanation
a) We have: 3.160 = 6.80 = 16.30 = 24.20, so 2 quantities x, y are two quantities in inverse proportion.
b) We have: 4.160 = 8. 80 = 320.20 \( \ne \)25.26 so 2 quantities x, y are not two quantities in inverse proportion.\(\dfrac{5}{{15}} = \dfrac {9}{{27}} = \dfrac{{15}}{{45}} = \dfrac{{24}}{{72}}\)
Solve lesson 6.24 page 18 Math 7 Textbook Connecting knowledge volume 2
Given that y is inversely proportional to x by the scaling factor a, x is inversely proportional to z by the scaling factor b. Ask y is directly or inversely proportional to z and what is the scaling factor?
Solution method
+ Use the definition of two quantities that are directly proportional and inversely proportional:
If y = ax (a is a non-zero constant), then y is proportional to x by the scaling factor a.
If \(y = \dfrac{a}{x}\)(a is a non-zero constant), then y is inversely 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.
If \(y = \dfrac{k}{z}\) ( k is a constant) then y and z are two quantities in inverse proportion.
Since y is inversely proportional to x by the scaling factor a, y = \(\dfrac{a}{x}\)
Since x is inversely proportional to z by the scaling factor b, x = \(\dfrac{b}{z}\)
Thus, \(y = \dfrac{a}{x} = \dfrac{a}{{\dfrac{b}{z}}} = a:\dfrac{b}{z} = a.\dfrac{ z}{b} = \dfrac{a}{b}.z\) ( \(\dfrac{a}{b}\) is constant because a,b are constants)
So y is proportional to z and the scaling factor is \(\dfrac{a}{b}\)
Detailed explanation
adsense
Since y is inversely proportional to x by the scaling factor a, y = \(\dfrac{a}{x}\)
Since x is inversely proportional to z by the scaling factor b, x = \(\dfrac{b}{z}\)
Thus, \(y = \dfrac{a}{x} = \dfrac{a}{{\dfrac{b}{z}}} = a:\dfrac{b}{z} = a.\dfrac{ z}{b} = \dfrac{a}{b}.z\) ( \(\dfrac{a}{b}\) is constant because a,b are constants)
So y is proportional to z and the scaling factor is \(\dfrac{a}{b}\).
Solve lesson 6.25 on page 18 Math 7 textbook Connecting knowledge volume 2
With the same amount of money to buy 17 sets of A4 grade 1 paper, how many sets of A4 grade 2 paper can buy, knowing that the price of class 2 banknotes is only 85% of the price of class 1 banknotes.
Solution method
The number of papers purchased and the corresponding price are two quantities that are inversely proportional.
Apply the property of two proportional quantities :\(\dfrac{{{x_1}}}{{{x_2}}} = \dfrac{{{y_2}}}{{{y_1}}}\)
Detailed explanation
Let the number of stacks of grade 2 paper that can be purchased be x (volume) (x > 0)
Since the amount of money does not change, the number of stacks of paper purchased and the corresponding price of money are inversely proportional.
Applying the property of two quantities that are inversely proportional, we have:
\(85\% = \dfrac{{17}}{x} \Rightarrow x = \dfrac{{17}}{{85\% }} = 20\)( satisfied)
So the number of second grade papers that can be bought is 20.
Solve lesson 6.26 page 18 Math 7 textbook Connecting knowledge volume 2
Three teams of tractors work on three fields of the same area. The first team completed the work in 4 days, the second team in 6 days and the third team in 8 days. How many plows does each team have, knowing that the number of plows in the first team is 2 more than the number of plows in the second team and the productivity of the machines is the same?
Solution method
Call the number of machines for each team respectively x,y,z (machine) (x,y,z \( \in \)N*).
The number of plows and the completion time are two quantities that are inversely proportional
Use the property of the series of equal ratios: \(\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = \dfrac{{a – c}}{ {b – d}}\)
Detailed explanation
Call the number of machines for each team respectively x,y,z (machine) (x,y,z \( \in \)N*).
Since the number of plows of the first team is 2 more than the number of plows of the second team, x – y = 2
Since the 3 fields have the same area and the same productivity of the machines, the number of plows and the completion time are inversely proportional.
Applying the property of two quantities that are inversely proportional, we have:
4x=6y=8z
\(\begin{array}{l} \Rightarrow \dfrac{x}{{\dfrac{1}{4}}} = \dfrac{y}{{\dfrac{1}{6}}} = \dfrac {z}{{\dfrac{1}{8}}} = \dfrac{{x – y}}{{\dfrac{1}{4} – \dfrac{1}{6}}} = \dfrac{ 2}{{\dfrac{1}{{12}}}} = 2:\dfrac{1}{{12}} = 2.12 = 24\\ \Rightarrow x = 24.\dfrac{1}{4} = 6\\y = 24.\dfrac{1}{6} = 4\\z = 24.\dfrac{1}{8} = 3\end{array}\)
So the number of machines in each team is 6 machines, 4 machines, and 3 machines respectively.