## Math 7 Chapter 2 Lesson 4: Some problems on quantities of inverse proportions

## 1. Summary of theory

To solve problems about quantities of inverse proportions, we apply the following knowledge:

\(\frac{{{y_1}}}{{{y_2}}} = \frac{{{x_1}}}{{{x_2}}}\).

\(\frac{{{y_1}}}{{{y_2}}} = \frac{{{x_2}}}{{{x_1}}} \Rightarrow \frac{{{y_1} + {y_2}}} {{{y_2}\,}} = \frac{{{x_2} + {x_1}}}{{{x_1}}},…\).

## 2. Illustrated exercise

**Question 1: **Two cars start at the same time and go towards each other from two provinces A and B 544km apart. Calculate how many kilometers away from A the two cars meet, knowing that the first car travels the whole distance AB in 12 hours and the second car takes 13:30 minutes.

**Solution guide:**

Let \({S_1},{V_1};{{\rm{S}}_2},{V_2}\) be the distance traveled and the velocities of the first and second cars traveling the same distance AB, respectively. speed is their travel time is inversely proportional to each other so we have \(\frac{{{V_1}}}{{{V_2}}} = \frac{{13,5}}{{12}} = \frac{9}{8}\) (1)

From the time of departure to the time of meeting, the two cars travel together in the same time, so the distance traveled and their speed are inversely proportional to each other. We have \(\frac{{{S_1}}}{{{S_2}}} = \frac{{{V_1}}}{{{V_2}}}\,\,\,\,(2)\)

From (1) and (2) we have \(\frac{{{S_1}}}{{{S_2}}} = \frac{9}{8}\)

So \(\frac{{{S_1}}}{9} = \frac{{{S_2}}}{8} = \frac{{{S_1} + {S_2}}}{{9 + 8}} = \frac{{544}}{{17}} = 32\)

So \({S_1} = 32.9 = 288\)

So the meeting point is 288km from A.

**Verse 2: **In a mechanical workshop, the master machinist completes a tool in 5 minutes, the assistant worker takes 9 minutes. If in the same time both work together, a total of 84 tools can be turned. Calculate the number of tools that each person has handy.

**Solution guide:**

Let x, y be the number of tools of the main worker and the auxiliary worker, respectively. We have the number of tools inversely proportional to the working time, so

\(\frac{x}{{\frac{1}{5}}} = \frac{y}{{\frac{1}{9}}}\) and x + y = 84

So \(\frac{x}{{\frac{1}{5}}} = \frac{y}{{\frac{1}{9}}} = \frac{{x + y}}{{ \frac{1}{5} + \frac{1}{9}}} = \frac{{84}}{{\frac{{14}}{{45}}}} = \frac{{84 – 45}}{{14}} = 270\)

So \(\begin{array}{l}\frac{x}{{\frac{1}{5}}} = 270 \Rightarrow x = \frac{1}{5}.270 = 54\\\frac {y}{{\frac{1}{9}}} = 270 \Rightarrow y = \frac{1}{9}.270 = 30\end{array}\).

The master craftsman made 54 tools.

The assistant worker can make 30 tools.

**Question 3: **Three units jointly built a bridge costing 340 million. The first unit has 8 vehicles and is located 1.5km from the bridge. The second unit has 4 vehicles and is located 3km from the bridge. The third unit has 6 vehicles and is located 1 km from the bridge.

Ask how much each unit must pay for the construction of the bridge, knowing that the amount to be paid is proportional to the number of vehicles and inversely proportional to the distance from the units to the bridge.

**Solution guide:**

Let x, y, z be the amount that each unit has to pay for the construction of the bridge (calculated in million dong).

We have: x + y + z = 340.

The amount to be paid is proportional to the number of cars on: x : y : z = 8 : 6 : 4

The amount to be paid is inversely proportional to the distance from each unit to the bridge, so:

\(x{\rm{ }}:{\rm{ }}y{\rm{ }}:{\rm{ }}z = \frac{1}{{1,5}}:\frac{1} {3}:1 = \frac{1}{3}:\frac{1}{3}:1\).

So \(\frac{x}{{\frac{{16}}{3}}} = \frac{y}{{\frac{6}{3}}} = \frac{z}{4} = \frac{{x + y + z}}{{\frac{{16}}{3} + \frac{6}{3} + 4}} = \frac{{x + y + z}}{ {\frac{{34}}{3}}} = \frac{{340}}{{\frac{{34}}{3}}} = 30\).

Hence: \(\begin{array}{l}x = \frac{{16}}{3}.30 = 160\\y = \frac{6}{3}.30 = 60\\z = 4.30 = 120\end{array}\).

So: The first unit pays 160 million, the second unit pays 60 million, and the third unit pays 120 million.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Divide the number 393 into parts that are inversely proportional to the numbers \(0,2;\,\,3\frac{1}{3};\,\,\frac{4}{5}\).

**Verse 2: **Prices are down 20%. How many % more goods can be purchased with the same amount of money?

**Question 3: **A person buys fabric to make three identical shirts. He bought three kinds of fabric 0.7m wide; 0.8m and 1.4m for a total fabric of 5.7m. Calculate the number of meters of fabric of each type he bought.

### 3.2. Multiple choice exercises

**Question 1: **Show that y is inversely proportional to x by the ratio k1 \(\left( {{k_1} \ne 0} \right)\) and x is inversely proportional to z by the ratio k2 \(\left( {{k_2} ) \ne 0} \right)\). Select the correct answers

A. y is z which is inversely proportional to each other by the scaling factor \(\frac{{{k_1}}}{{{k_2}}}\)

B. y is z which is inversely proportional to each other by the scaling factor \(\frac{{{k_2}}}{{{k_1}}}\)

C. y is z proportional to each other by the scaling factor k1.k2

D. y is z proportional to each other by the scaling factor \(\frac{{{k_1}}}{{{k_2}}}\)

**Verse 2: **A car traveling from A to B with a speed of 50km/h takes 2 hours and 15 minutes. How long does it take for a car to travel from A to B at 45km/hr?

A. 3.25 hours

B. 3, 5 hours

C. 3 hours

D. 2.5 hours

**Question 3: **If a person travels from A to B by bicycle in 90 minutes, then he or she travels from B to A by motorbike at twice the speed of bicycle, then it takes:

A. 45 minutes

B. 180 minutes

C. 60 minutes

D. 30 minutes

**Question 4: **A car traveling from A to B at a speed of 50 km/h takes 6 hours. How long does it take to get from B to A with a speed of 30 km/h?

A. 9.6 hours

B. 7 o’clock

C. 10 o’clock

D. 8 o’clock

**Question 5: **Suppose 5 workers complete a job in 16 hours. Ask 8 workers (with the same productivity) to complete the work in how many hours?

A. 10 o’clock

B. 12 o’clock

C. 13 hours

D. 15 hours

## 4. Conclusion

Through the lecture Some problems on quantities of this inverse proportion, students need to recognize and do problems related to quantities of inverse proportion.

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