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

### 1.1. Knowledge to remember

**Problem 1:** The difference of two numbers is 24. The ratio of the two numbers \(\frac{3}{5}\). Find two of them.

*Solution*

We have a diagram:

According to the diagram, the equal part difference is:

5 – 3 = 2 (parts)

The small number is :

24 : 2 × 3 = 36

The big number is:

36 + 24 = 60

Answer: Number of children : 36 ;

Big number : 60.

**Problem 2: ** A rectangle is 12 meters longer than it is wide. Find the length and width of the figure, given that the length is 7474 the width.

*Solution*

We have a diagram:

According to the diagram, the equal part difference is:

7 – 4 = 3 (parts)

The length of the rectangle is:

12 : 3 × 7 = 28 (m)

The width of the rectangle is:

28 – 12 = 16 (m)

Answer: Length: 28m;

Width: 16m

### 1.2. Textbook exercise solution page 151

**Lesson 1: **The first number is 123 less than the second. The ratio of the two numbers is \(\frac{2}{5}\). Find two of them.

**Solution guide:**

- Step 1: Draw a diagram: Considering the small number consists of 5 equal parts, the large number has 9 such parts.
- Step 2: Find the difference of equal parts
- Step 3: Find the value of a part by dividing the difference of two numbers by the difference of the 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 big number (take the small number plus the difference of two numbers…)

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

*Solution*

We have a diagram:

According to the diagram, the equal part difference is:

5 – 2 = 3 (parts)

The first number is:

123 : 3 x 2 = 82

The second number is:

82 + 123 = 205

Answer: First number: 82;

Second number: 205.

**Lesson 2: **My mother is 25 years older than me. Child’s age is equal to \(\frac{2}{7}\) mother’s age. Calculate the age of each person.

__Solution guide:__

- Step 1: Draw a diagram: Considering the child’s age (as the number of children) consists of 2 equal parts, the mother’s age (as the large number \) consists of 7 such parts.
- Step 2: Find the difference of equal parts.
- Step 3: Find the value of a part by dividing the difference of two numbers by the difference of the 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 big number (take the small number plus the difference of two numbers…)

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

*Solution*

We have a diagram:

According to the diagram, we have equal part difference as:

7 – 2 = 5 (parts)

Your child’s age is:

25: 5 x 2 = 10 (age)

Mother’s age is:

10 + 25 = 35 (age)

Answer: Mother: 35 years old;

Child: 10 years old

**Lesson 3:** The difference of two numbers is equal to the smallest three-digit number. The ratio of those two numbers is \(\frac{9}{5}\). Find two of them.

__Solution guide:__

- Step 1: Find the difference of two numbers: The smallest three-digit number is 100. So the difference of the two numbers to find is 100.
- Step 2: Draw a diagram: Considering the small number consists of 5 equal parts, the large number has 9 such parts.
- Step 3: Find the difference of equal parts.
- Step 4: Find the value of a fraction by dividing the difference of two numbers by the difference of the 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 big number (take the small number plus the difference of two numbers…).

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

The smallest three-digit number is 100. So the difference between the two numbers to find is 100.

*Solution*

We have a diagram:

According to the diagram, the equal part difference is:

9 – 5 = 4 (parts)

The small number is :

100 : 4 x 5 = 125

The big number is:

125 + 100 = 225

Answer: Number of children: 125;

Large number: 225.

### 1.3. Solve the exercises Textbook Practice page 151 – Period 1

**Lesson 1: **The difference of two numbers is 85. The ratio of two numbers is \(\frac{3}{8}\). Find two of them.

__Solution guide:__

- Step 1: Draw a diagram: Considering the small number consists of 3 equal parts, the large number has 8 such parts.
- Step 2: Find the difference of equal parts.
- Step 3: Find the value of a part by dividing the difference of two numbers by the difference of the 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 big number (take the small number plus the difference of two numbers…).

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

*Solution*

We have a diagram:

According to the diagram, we have the equal part difference as:

8−3 = 5 (parts)

The small number is:

85 : 5 × 3 = 51

The big number is:

51 + 85 = 136

Answer: Number of children: 51;

Big number: 136.

**Lesson 2: **The number of colored light bulbs is 250 more than the number of white light bulbs. Find the number of bulbs of each type, knowing that the ratio of colored bulbs is \(\frac{5}{3}\) the number of white bulbs.

__Solution guide:__

- Step 1: Draw a diagram: Considering the number of white light bulbs (as a small number) consists of 3 equal parts, the number of colored light bulbs (as a large number) consists of 5 such parts.
- Step 2: Find the difference 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 big number (take the small number plus the difference of two numbers…)

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

*Solution*

We have a diagram:

According to the diagram, we have equal part difference as:

5 − 3 = 2 (parts)

The number of white bulbs is:

250 : 2 × 3 = 375 (light bulb)

Number of color bulbs is:

375 + 250 = 625 (light bulb)

Answer: 375 white light bulbs;

625 color bulbs.

**Lesson 3: **Class 4A has 35 students and class 4B has 33 students participating in planting trees. Class 4A planted 10 trees more than class 4B. Ask how many trees each class can plant, knowing that each student planted the same number of trees.

__Solution guide:__

- Find the student difference of the two classes.
- Find the number of trees each student can plant = difference of trees of two classes : difference of students.
- Number of trees planted by class 4A = number of trees each student can plant x number of students in class 4A.
- Number of trees planted in class 4B = number of trees planted in class 4A −10 trees.

*Solution*

The number of students in class 4A is more than the number of students in class 4B:

35−33 = 2 (students)

The number of trees each student planted is:

10 : 2 = 5 (tree)

The number of trees planted in class 4A is:

5 × 35 = 175 (tree)

The number of trees planted by class 4B is:

175 − 10 = 165 (tree)

Answer: Class 4A: 175 trees;

Class 4B: 165 trees.

**Lesson 4: **State the problem and solve it according to the following diagram

**Solution guide:**

- Look at the diagram to find the ratio and difference of two numbers, then solve the appropriate problem.
- Do the maths :

Step 1: Find the difference of equal parts

Step 2: Find the value of a part by dividing the difference of two numbers by the difference of the 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 big number (take the small number plus the difference of two numbers…)

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

*Solution :*

The problem can be stated as follows:

Two numbers equal 72. The ratio of the two numbers is \(\frac{5}{9}\). Find two of them.

*Solution*

According to the diagram, the equal part difference is:

9 − 5 = 4 (parts)

The small number is:

72 : 4 × 5 = 90

The big number is:

90 + 72 = 162

Answer: Number of children: 90;

Large numbers: 162.

### 1.4. Solve the exercises Textbook Practice page 151 – Period 2

**Lesson 1: **The difference of two numbers is 30. The first number is 3 times the second. Find two of them.

__Solution guide:__

- Step 1: Draw a diagram: Considering the second number (as a small number) consists of 1 part, the first number (as a large number) consists of 3 such parts.
- Step 2: Find the difference 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 big number (take the small number plus the difference of two numbers…).

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

*Solution*

We have a diagram:

According to the diagram, the equal part difference is:

3−1 = 2 (parts)

The second number is:

30 : 2 × 1 = 15

The first number is:

15 × 3 = 45

Answer: First number: 45;

Second number: 15.

**Lesson 2: **The second number is 60 more than the first number. If the first number is increased 5 times, the second number is obtained. Find two of them.

__Solution guide:__

- Step 1: Draw a diagram: If the first number is 5 times higher, the second number is obtained, so the second number is 5 times the first number. Considering the first number (as a small number) consists of 1 part, the second number (as a large number) consists of 5 such parts.
- Step 2: Find the difference 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 big number (take the small number plus the difference of two numbers…).

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

*Solution*

If the first number is multiplied 5 times, the second number is obtained, so the second number is 5 times the first number.

We have a diagram:

According to the diagram, we have the equal part difference as:

5−1 = 4 (parts)

The first number is:

60 : 4 × 1 = 15

The second number is:

15 × 5 = 75

Answer: First number: 15;

Second number: 75.

**Lesson 3: **A shop has 540kg less sticky rice than plain rice. Calculate the number of rice of each type, knowing that the number of sticky rice is equal to \(\frac{1}{4}\) the number of plain rice.

__Solution guide:__

- Step 1: Draw a diagram: Considering the number of sticky rice (as a small number) includes 1 part, the number of sticky rice (as a large number) includes 4 such parts.
- Step 2: Find the difference of equal parts.
- Step 5: Find the big number (take the small number plus the difference of two numbers…).

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

*Solution*

We have a diagram:

According to the diagram, the equal part difference is:

4−1 = 3 (parts)

The number of glutinous rice is:

540 : 3 × 1 = 180(kg)

The number of plain rice is:

180 + 540 = 720(kg)

Answer: Sticky rice: 180kg;

Plain rice: 720kg.

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

__Solution guide:__

- Look at the diagram to find the ratio and difference of two numbers, then solve the appropriate problem.
- Do the maths :

Step 1: Find the difference of equal parts.

Step 2: Find the value of a part by dividing the difference of two numbers by the difference of the 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 big number (take the small number plus the difference of two numbers…).

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

*Solution :*

The problem can be stated as follows:

The number of orange trees in the garden is equal to \(\frac{1}{6}\) the number of pineapple trees, knowing the number of pineapple trees is 170 more than the number of orange trees. How many orange trees are there in the garden? How many pineapples?

*Solution*

According to the diagram, the equal part difference is:

6−1 = 5 (parts)

The number of orange trees is:

170 : 5 × 1 = 34 (tree)

The number of pineapple trees is:

34 + 170 = 204 (tree)

Answer: Orange trees: 34 trees;

Pineapple tree: 204 trees

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