Multiply a number by a difference
1.1. Knowledge to remember
Calculate and compare the values of two expressions:
3 × (7−5) and 3 × 7−3 × 5
We have: 3 × (7−5) = 3 × 2 = 6
3 × 7−3 × 5 = 21−15 = 6
So: 3 × (7−5) = 3 × 7−3 × 5.
When we multiply a number by a difference, we can multiply that number by a minus and minus number, respectively, and then subtract the two results together.
a × (b−c) = a × b−a × c
1.2. Solving Textbook Exercises
Lesson 1: Calculate the value of the expression and then write it in the blank (according to the pattern)
a 
b 
c 
ax (b – c) 
axb – axc 
3 
7 
3 
3 x (7 – 3) = 12 
3 x 7 – 3 x 3 = 12 
6 
9 
5 


8 
5 
2 


Solution guide:
 Replace letters with numbers and then evaluate those expressions.
a 
b 
c 
ax (b – c) 
axb – axc 
3 
7 
3 
3 x (7 – 3) = 12 
3 x 7 – 3 x 3 = 12 
6 
9 
5 
6 x (9 – 5) = 24 
6 x 9 – 6 x 5 = 24 
8 
5 
2 
8 x (5 – 2) = 24 
8 x 5 – 5 x 2 = 24 
Lesson 2: Apply the property to multiply a number by a difference to calculate (in the form)
Template: 26 x 9 = 26 x (10 – 1)
= 26 x 10 – 26 x 1
= 260 – 26 = 234
a) 47 x 9 b) 138 x 9
24 x 99 123 x 99
Split 9 = 10 – 1, 99 = 100 – 1, then apply multiplying a number by a difference to calculate the expression value.
a) 47 x 9 = 47 x (10 – 1)
= 47 x 10 – 47 x 1
= 470 – 47 = 423
24 x 99 = 24 x ( 100 – 1)
= 24 x 100 – 24 x 1
= 2400 – 24 = 2376
b) 138 x 9 = 138 x (10 – 1)
= 138 x 10 – 138 x 1
= 1380 – 138 = 1242
123 x 99 = 123 x (100 – 1)
= 123 x 100 – 123 x 1
= 12300 – 123 = 12177
Lesson 3: An egg store has 40 egg racks, each with 175 eggs. The store sold out of 10 eggs. How many eggs are left in that store?
Solution guide:
 To calculate the remaining egg price, we subtract the original egg price from the sold egg price.
 To find the number of eggs left, multiply the number of eggs in one rack by the number of eggs left.
Solution
The remaining egg racks are:
40 – 10 = 30 (price)
The store’s remaining eggs are:
175 x 30 = 5250 (egg)
Answer: 5250 fruits.
Lesson 4: Calculate and compare the value of two expressions
( 7 – 5) × 3 and 7 × 3 – 5 × 3
From the results compare and show how to multiply a difference by a number.
Solution guide:
 Expressions with parentheses are evaluated in parentheses first, outside parentheses.
 For expressions with multiplication and addition, we calculate the multiplication first and the addition later.
We have: (7 – 5) × 3 = 2 × 3 = 6
7 × 3 – 5 × 3 = 21 – 15 = 6
So the two expressions are already equal, or :
(7 – 5) × 3 = 7 × 3 – 5 × 3
When multiplying a difference by a number, we can multiply the subtracted number, the subtracted number by that number, and then subtract the two results together.
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