The Effect of Addition and Multiplication on the Order of Numbers

Peter’s schoolbag is heavier than Marie’s. Deciding that their bags are too heavy, each of them takes out the third-year mathematics book. It is obvious that Peter’s bag will still be the heavier of the two.

What is the mathematical translation of this situation?

I. Addition and order

1. Property

Example: We want to compare x = 23.4 + 7.986 and y = 19.74 + 7.986.
Carrying out the two additions is one way to find the answer, but there is another way as well. Effectively we are adding the same number, 7.986, to both 23.4 and 19.74. Since 23.4 > 19.74, we see that x > y.

Rule: If a, b, and c are integers, the integers a + c and b + c come in the same order as a and b.
In other words, if we take the same number and add it to (or subtract it from) each member of an inequality, we obtain an inequality in the same direction, that is equivalent to the original.

2. Applications

Example 1: We want to compare $\frac{3}{7}-\frac{9}{11}$ and $\frac{3}{13}-\frac{9}{11}$ .

We note that we are subtracting the same number, $\frac{9}{11}$ , from both $\frac{3}{7}$ and $\frac{3}{13}$ . We know that $\frac{3}{7}>\frac{{\displaystyle 3}}{{\displaystyle 13}}$, since two numbers with the same positive numerator are arranged in the inverse order of their denominators.

Therefore we know that $\frac{{\displaystyle 3}}{{\displaystyle 7}}-\frac{9}{11}>\frac{{\displaystyle 3}}{{\displaystyle 13}}-\frac{{\displaystyle 3}}{{\displaystyle 11}}$.


Example 2: X is an integer such that X + 3 ≤ -7. What does this tell us about X?


If we subtract the same number from each member of an inequality, we find an inequality in the same direction.
Therefore we know that X + 3-3 ≤ -7-3, that is, X ≤ -10. So the number X is less than or equal to -10.

II. Multiplication and order

1. Property

Examples: We want to compare x = 4.7 × 2.93 and y = 7.9 × 2.93.


Effectively we have multiplied 4.7 and 7.9 by the same positive number, 2.93. Since 4.7 < 7.9, we know that x < y.
In the same way, (-3.5) × 14 < (-2.9) × 14, since -3.5 < -2.9.


Rule: If a, b, and c are integers, where c is always positive, the integers a × c and b × c come in the same order as a and b.
In other words, if we multiply or divide each member of an inequality by the same positive number, we obtain an inequality in the same direction, which is equivalent to the original.


Note: What happens if we multiply each member by a negative number? This gives an inequality equivalent to the original in the opposite direction.


For example, 4 < 7 and (-3) × 4 > (-3) × 7 (that is, -12 > -21).
This example shows the importance of the “strictly positive” condition.

2. Applications

Example 1: We want to compare $\frac{{\displaystyle 3}}{{\displaystyle 11}}\times \frac{4}{7}$ and $\frac{{\displaystyle 5}}{{\displaystyle 11}}\times \frac{{\displaystyle 4}}{{\displaystyle 7}}$.

We note that we are multiplying both $\frac{{\displaystyle 3}}{{\displaystyle 11}}$ and $\frac{{\displaystyle 5}}{{\displaystyle 11}}$ by the same positive number, $\frac{{\displaystyle 4}}{{\displaystyle 7}}$.


Since $\frac{{\displaystyle 3}}{{\displaystyle 11}}<\frac{{\displaystyle 5}}{{\displaystyle 11}}$, we can see that $\frac{{\displaystyle 3}}{{\displaystyle 11}}\times \frac{{\displaystyle 4}}{{\displaystyle 7}}<\frac{{\displaystyle 5}}{{\displaystyle 11}}\times \frac{{\displaystyle 4}}{{\displaystyle 7}}$.


Example 2: x is an integer such that 5 x ≤-$\frac{{\displaystyle 4}}{{\displaystyle 7}}$. What does this tell us about x?

If we divide each member of an inequality by the same positive real number, this gives an equivalent inequality in the same direction. Therefore $\frac{{\displaystyle 1}}{{\displaystyle 5}}\times 5\times$ ≤ $\frac{{\displaystyle 1}}{{\displaystyle 5}}\times \left(-\frac{{\displaystyle 4}}{{\displaystyle 7}}\right)$, that is, x ≤ $-\frac{{\displaystyle 4}}{{\displaystyle 35}}$.
So the number x is less than or equal to $-\frac{{\displaystyle 4}}{{\displaystyle 35}}$.

Read more:
Reading and Writing Decimal Numbers
Multiplying or Dividing a Decimal Number by 10, 100, or 1,000
Multiplying Decimal Numbers
Dividing Decimal Numbers
Converting Decimals to Fractions
Describing Different Types of Numbers
Dividing Whole Numbers with a Remainder
Comparing and Ordering Decimal Numbers
Adding and Subtracting Decimal Numbers