Find two numbers when the sum and ratio of the two numbers are known

### 1.1. Knowledge to remember

**Problem 1:** The sum of two numbers is 96. The ratio of the two numbers \(\frac{3}{5}\). Find two of them.

*Solution*

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is :

3 + 5 = 8 (parts)

The small number is :

96 : 8 × 3 = 36

The big number is:

96 – 36 = 60

* Answer : *Number of children : 36 ;

Big number : 60.

**Problem 2: **Minh and Khoi have 25 notebooks. The number of Minh’s notebooks is equal to \(\frac{2}{3}\) the number of Khoi’s notebooks. How many notebooks does each person has ?

*Solution*

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is :

2 + 3 = 5 (parts)

Minh’s notebooks are:

25 : 5 × 2 = 10 (book)

The number of Khoi’s notebooks is:

25 – 10 = 15 (book)

* Answer :* Minh: 10 notebooks;

Khoi: 15 notebooks

### 1.2. Textbook exercise solution page 148

**Lesson 1:** The sum of two numbers is 333. The ratio of the two numbers is \(\frac{2}{7}\). Find two of them.

__Solution guide:__

- Step 1: Draw a diagram: considering the small number consists of 2 equal parts, the large number has 7 such parts.
- Step 2: Find the total number of equal parts.
- Step 3: Find the value of 1 part by dividing the sum of two numbers by the total number of equal parts.
- Step 4: Find the small number (multiply the partial value by the number of parts of the smaller number).
- Step 5: Find the large number (multiply the partial value by the number of parts of the large number).

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is :

2 + 7 = 9 (parts)

Small number is: 333 : 9 x 2 = 74

Big number is: 333 – 74 = 259

Answer: Large number: 259 ;

Number of children: 74.

**Lesson 2: **Two warehouses hold 125 tons of paddy, in which the number of paddy in the first warehouse is equal to \(\frac{3}{2}\) the number of paddy in the second. How many tons of rice does each warehouse hold?

__Solution guide:__

- Step 1: Draw a diagram: considering the number of paddy in the second warehouse (playing the role of small number) consists of 2 equal parts, the number of paddy in the first warehouse (playing the role of a large number) consists of 3 such parts.
- Step 2: Find the total number of equal parts.
- Step 3: Find the value of 1 part by dividing the sum of two numbers by the total number of equal parts.
- Step 4: Find the small number (multiply the partial value by the number of parts of the smaller number).
- Step 5: Find the large number (multiply the partial value by the number of parts of the large number).

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram the total number of equal parts is:

3 + 2 = 5 (parts)

The number of paddy in the first barn is:

125 : 5 x 3 = 75 (tons)

The number of paddy in the second barn is:

125 – 75 = 50 (tons)

Answer: First warehouse: 75 tons;

Second warehouse: 50 tons.

**Lesson 3: **The sum of two numbers is equal to the largest two-digit number. The ratio of those two numbers is \(\frac{4}{5}\). Find two of them.

__Solution guide:__

- Step 1: Find the sum of two numbers: The largest two-digit number is 99. So the sum of the two numbers is 99.
- Step 2: Draw a diagram: considering the small number consists of 4 equal parts, the large number has 5 such parts.
- Step 3: Find the total number of equal parts.
- Step 4: Find the value of 1 part by dividing the sum of two numbers by the total number of equal parts.
- Step 5: Find the small number (multiply the partial value by the number of parts of the smaller number).
- Step 6: Find the large number (multiply the partial value by the number of parts of the large number).

*Attention *: Step 4 and step 5 can be combined into one step.

The largest two-digit number is 99. So the sum of the two numbers is 99.

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is :

4 + 5 = 9 (parts)

Small number is : 99 : 9 x 4 = 44

Big number is : 99 – 44 = 55

Answer : Large number : 55 ;

Number of children: 44.

### 1.3. Solve the exercises Textbook Practice page 148

**Lesson 1: **Find two numbers, knowing their sum is 198 and their ratio is \(\frac{3}{8}\).

__Solution guide:__

- Step 1: Draw a diagram: considering the small number consists of 3 equal parts, then the large number has 8 such parts.
- Step 2: Find the total number of equal parts.
- Step 3: Find the value of 1 part by dividing the sum of two numbers by the total number of equal parts.
- Step 4: Find the small number (multiply the partial value by the number of parts of the smaller number).
- Step 5: Find the large number (multiply the partial value by the number of parts of the large number).

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is :

3 + 8 = 11 (parts)

The small number is :

198 : 11 x 3 = 54

The big number is:

198 – 54 = 144

Answer: Number of children: 54 ;

Large numbers: 144.

**Lesson 2: **A person has sold 280 oranges and tangerines, where the number of oranges is equal to \(\frac{2}{5}\) the number of tangerines. Find the number of oranges and tangerines sold.

__Solution guide:__

- Step 1: Draw a diagram: considering the number of oranges (as a small number) consists of 2 equal parts, the number of tangerines (as a large number) consists of 5 such parts.
- Step 2: Find the total number of equal parts.

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is:

2 + 5 = 7 (parts)

The number of oranges sold is:

280 : 7 x 2 = 80 (fruit)

The number of tangerines sold is:

280 – 80 = 200 (fruit)

Answer: Number of oranges: 80;

Number of tangerines: 200.

**Lesson 3: **Class 4A and class 4B planted 330 trees. Class 4A has 34 students, class 4B has 32 students. Ask how many trees each class can plant, knowing that each student planted the same number of trees.

__Solution guide:__

- Step 1: Find the total number of students in the two classes.
- Step 2: Find the number of trees each student can plant by dividing the total number of trees by the total number of students.
- Step 3: Find the number of trees that class 4A can plant, multiply the number of trees each student grows by the number of students in class 4A.
- Step 4: Find the number of trees planted in class 4B, we take the total number of trees planted in class 4A minus the number of trees in class 4A.

*Solution*

The total number of students in the two classes is:

34 + 32 = 66 (students)

The number of trees each student planted is:

330 : 66 = 5 (tree)

The number of trees planted in class 4A is:

5 x 34 = 170 (tree)

The number of trees planted by class 4B is:

330 – 170 = 160 (tree)

Answer: Class 4A: 170 trees;

Class 4B: 160 trees.

**Lesson 4:** A rectangle has perimeter 350m, width equals \(\frac{3}{4}\) length. Find the length and width of that rectangle.

__Solution guide:__

- Step 1: Find half circumference = perimeter : 2
- Step 2: Draw a diagram: considering the width (playing the role of a small number) consists of 3 equal parts, the length (playing the role of a large number) consists of 4 such parts.
- Step 3: Find the total number of equal parts.
- Step 4: Find the value of 1 part by dividing the sum of two numbers by the total number of equal parts.
- Step 5: Find the small number (multiply the partial value by the number of parts of the smaller number).
- Step 6: Find the large number (multiply the partial value by the number of parts of the large number).

*Attention :* Step 4 and step 5 can be combined into one step.

The half perimeter of the rectangle is:

350 : 2 = 175 (m)

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is :

3 + 4 = 7 (parts)

The width of the rectangle is:

175 : 7 x 3 = 75 (m)

The length of the rectangle is:

175 – 75 = 100 (m).

Answer: Length: 100m ;

Width: 75m.

### 1.4. Solve the exercises Textbook Practice page 149

**Lesson 1: **A 28m long rope is cut into two pieces, the first is three times as long as the second. How many meters long is each segment?

__Solution guide:__

- Step 1: Draw a diagram: considering the second segment (as a small number) consists of 1 part, the first segment (as a large number) consists of 3 such parts.
- Step 2: Find the total number of equal parts.

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram, the total number of parts is equal:

3 + 1 = 4 (parts)

The second segment in meters is:

28 : 4 x 1 = 7 (m)

The first segment is meters long:

28 – 7 = 21 (m)

Answer: First section: 21m;

Second segment: 7m.

**Lesson 2: **A group of students has 12 students, in which the number of boys is half of the number of girls. How many boyfriends and girlfriends does that group have?

__Solution guide:__

- Step 1: Draw a diagram: considering the number of boyfriends (the role is the small number) consists of 1 part, the number of girls (the role is the large number) includes 2 such parts.
- Step 2: Find the total number of equal parts.

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is:

1+2 = 3 (parts)

Boyfriends are:

12 : 3 = 4 (you)

The number of girlfriends is:

12 – 4 = 8 (you)

Answer: Boyfriend: 4 friends;

Girlfriend: 8 friends.

**Lesson 3: **The sum of the two numbers is 72. Find the two numbers, knowing that if the large number is reduced by 5 times, then the small number is obtained.

__Solution guide:__

- Step 1: Draw a diagram: considering the small number consists of 1 part, the large number consists of 5 such parts.
- Step 2: Find the total number of equal parts.

*Attention :* Step 3 and step 4 can be combined into one step.

We have a diagram:

*Solution*

According to the diagram, the total number of equal parts is:

5 + 1 = 6 (parts)

The small number is:

72 : 6 x 1 = 12

The big number is:

72 – 12 = 60

Answer: Number of children: 12;

Large number: 60.

**Lesson 4:** State the problem, then solve the problem according to the following diagram

__Solution guide:__

- Use the diagram to state the appropriate problem.
- Do the maths :

Step 1: Find the total number of equal parts.

Step 2: Find the value of 1 part by dividing the sum of two numbers by the total number of equal parts.

Step 3: Find the small number (multiply the partial value by the number of parts of the smaller number).

Step 4: Find the large number (multiply the partial value by the number of parts of the large number).

*Attention :* Step 2 and step 3 can be combined into one step.

The problem can be formulated according to the following scheme:

There are two oil tanks containing a total of 180l, the first oil tank has the same amount of oil as \(\frac{1}{4}\) the second oil tank. How many liters of oil does each barrel contain?

*Solution*

According to the diagram, the total number of equal parts is:

1 + 4 = 5 (parts)

The number of liters of oil contained in the first tank is:

180 : 5 x 1 = 36 (liters)

The number of liters of oil contained in the second tank is:

180 – 36 = 144 (liters)

Answer: First barrel: 36 liters;

Second barrel: 144 liters.

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