Consecutive integers problems
1.The sum of two consecutive integers is 120. Find the value of the smaller integer.
Answer and Explanation:
The first number is 59, the second number is 59 + 2 = 61.
2.The sum of two consecutive odd integers is 60. What are the integers?
Answer and Explanation:
The two consecutive numbers are, x = 29. x+2 = 29 + 2 = 31.
3.What is the sum of the first 60 consecutive odd numbers?
Answer and Explanation:
We have to find the sum of the first 60 consecutive off numbers. So . Therefore, the required sum = 60 2 = 3600 .
4. What are five consecutive integers whose sum is 60?
Answer and Explanation:
Here, 60/5 = 12. So, the 5 consecutive numbers are 10, 11, 12, 13 and 14.12
5. What is the sum of the first n consecutive odd integers?
Answer and Explanation:
sum of the first n odd numbers is n x n or n2. For example, if we put n = 21, then we have 21 x 21 = 441, which is equal to the sum of the first 21 odd numbers.
Facts
The term consecutive numbers is used to frame word problems.
The sum of any two consecutive numbers is always odd. Example, 5 + 6 = 11

Odd Consecutive Integer Formula In mathematics, we represent an odd integer as 2n + 1. If 2n + 1 is an odd integer, (2n + 3) and (2n + 5) will be the next two odd consecutive integers. For example, let 2n + 1 be 7, which is an odd integer. We find its consecutive integers as (7 + 2) and (7 + 4), or 9 and 11.

When you start counting natural numbers you are just counting the consecutive numbers or consecutive integers. Consecutive integers are integers that follow each other in a fixed sequence. Did you know that whenever you number items you are using Consecutive Integers? In fact, whenever you count by ones from any number in a set you obtain Consecutive Integers. Consecutive integers are integers that follow in a fixed sequence, each number being 1 more than the previous number, Consecutive integers are represented by n, n +1, n + 2, n + 3, …, where n is an integer.

Examples of Consecutive Integer

For example: 23, 24, 25

Look at the following. The first set is called consecutive positive integers and the second set is called consecutive negative integers.

Example 1: 1, 2, 3, 4, 5…..

Example 2: -1, -2, -3, -4, -5, -6,…..

In the first example a set of consecutive integers is found by adding 1 to 0. You can represent the first set with this expression: n + 1, with n = 0, 1, 2, …..

The second set of consecutive integers is found by subtracting 1 from 0. You can represent the second set with this expression: 1 − n, with n = 2, 3, 4, 5,…..

Type of Consecutive Integers

There are mainly three types of consecutive integers:

1.Normal consecutive integers (2, 3, 4, 5, ……)

2.Even consecutive integers (2, 4, 6, 8, ……..)

3.Odd consecutive integers (3, 5, 7, 9, ………)

1.Even Consecutive Integers

Consecutive even integers are the set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is divisible by 2.

Consecutive even integers are even integers that follow each other by a difference of 2. If x is an even integer, then x + 2, x + 4, x + 6 and x + 8 are consecutive even integers.

Examples:

4, 6, 8, 10, …

-6, -4, -2, 0, …

124, 126, 128, 130, ..

You can represent consecutive even integers with the following expression: 2n + 2 with n = 0, 1, 2, 3….

2.Odd Consecutive Integers

Consecutive odd integers are the set of integers such that each integer in the set differs from the previous integer by a difference of 2 and each integer is an odd number.

Consecutive odd integers are odd integers that follow each other by the difference of 2. If x is an odd integer, then x + 2, x + 4, x + 6 and x + 8 are consecutive odd integers.

Examples:

1,3, 5, 7, 9, 11,…

-7, -5, -3, -1, 1,…

-25, -23, -21,….

You can represent consecutive odd integers with the following expression: 2n + 1 with n = 0, 1, 2, 3….

Odd Consecutive Integer Formula In mathematics, we represent an odd integer as 2n + 1. If 2n + 1 is an odd integer, (2n + 3) and (2n + 5) will be the next two odd consecutive integers. For example, let 2n + 1 be 7, which is an odd integer. We find its consecutive integers as (7 + 2) and (7 + 4), or 9 and 11.

When you start counting natural numbers you are just counting the consecutive numbers or consecutive integers. Consecutive integers are integers that follow each other in a fixed sequence. Did you know that whenever you number items you are using Consecutive Integers? In fact, whenever you count by ones from any number in a set you obtain Consecutive Integers. Consecutive integers are integers that follow in a fixed sequence, each number being 1 more than the previous number, Consecutive integers are represented by n, n +1, n + 2, n + 3, …, where n is an integer.

Examples of Consecutive Integer

For example: 23, 24, 25

Look at the following. The first set is called consecutive positive integers and the second set is called consecutive negative integers.

Example 1: 1, 2, 3, 4, 5…..

Example 2: -1, -2, -3, -4, -5, -6,…..

In the first example a set of consecutive integers is found by adding 1 to 0. You can represent the first set with this expression: n + 1, with n = 0, 1, 2, …..

The second set of consecutive integers is found by subtracting 1 from 0. You can represent the second set with this expression: 1 − n, with n = 2, 3, 4, 5,…..

Percentage Formula

To determine the percentage, we have to divide the value by the total value and then multiply the resultant by 100.

Percentage formula = (Value/Total value) × 100

Example: 2/5 × 100 = 0.4 × 100 = 40 per cent

How to calculate the percentage of a number?

To calculate the percentage of a number, we need to use a different formula such as:

P% of Number = X

where X is the required percentage.

If we remove the % sign, then we need to express the above formulas as;

P/100 * Number = X

Example: Calculate 10% of 80.

Let 10% of 80 = X

10/100 * 80 = X

X = 8

percentage

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

Example -1

A bakery had 252 customers last week.

The customers increased to 378 in this week.

To find:

Percentage of increase in customers.

Explanation:

From the question, we have

Percentage increase =

Example -2

An alloy contains 26 % of copper. What quantity of alloy is required to get 260 g of copper?

Solution:

Let the quantity of alloy required = m g

Then 26 % of m =260 g

⇒ 26/100 × m = 260 g

⇒ m = (260 × 100)/26 g

⇒ m = 26000/26 g

⇒ m = 1000 g

Examples of percentages are:

10% is equal to 1/10 fraction

20% is equivalent to ⅕ fraction

25% is equivalent to ¼ fraction

50% is equivalent to ½ fraction

75% is equivalent to ¾ fraction

90% is equivalent to 9/10 fraction

Place Value and Face Value

The face value of a digit in any number is the digit itself. Whether the number is single-digit, double-digit, or any number, each digit has its face value. Let us understand this using the following examples.

• If 4 is the given number, the face value of 4 is 4, and the place value of 4 is also 4 (4 ones = 4 × 1 = 4).
• For a given number 78, the face value of 7 is 7 and its place value is 70 (7 tens = 7 × 10 = 70).
• For 52369, the face value of 3 is 3 while its place value is 300 (3 hundreds = 3 × 100 = 300).

Difference between Place Value and Face Value

Place value describes the position of a digit in a given number. On the other hand, face value represents the number itself.

Let us take an example of a number say, 1437. The table given below explains the difference between the place value and the face value of digits in this number.

Place Value Examples

Example 1: Write the place value of the underlined digit: 645

Solution:

Since 6 is in the hundreds place, the place value of 6 in 645 is 6 hundreds. This means the place value of 6 is expressed as 6 × 100 = 600.

Example 2: A number has 4 thousand, 7 hundreds, and 8 tens. What is the number?

Solution:

The place value of the following digits are:

4 thousands = 4,000

7 hundreds = 700

8 tens = 80

Adding these numbers together, we get: 4,000 + 700 + 80 = 4780. Therefore, the number is 4780.

Place Value Chart with Decimals

The decimal place value chart shows the place value of the digits in a decimal number. A decimal number system is used to express the whole numbers and fractions together using a decimal point. This decimal point lies between the whole number part and the fractional part. While the whole number part follows the usual place value chart of ones, tens, hundreds, and so on, there is a slight difference in the place value of the numbers to the right of the decimal point. If we go to the right after the decimal, the place values start from tenths and go on as hundredths, thousandths, and so on. The first place to the right of the decimal is on the one-tenth (1/10th) position, the next one is 1/100 and it goes on. Observe the following place value chart for decimal numbers.

Place Value Definition

Place value is the value of a digit according to its position in the number such as ones, tens, hundreds, and so on. For example, the place value of 5 in 3458 is 5 tens, or 50. However, the place value of 5 in 5781 is expressed as 5 thousand or 5,000. It is important to understand that a digit can be the same, but its value depends on its position in the number.

Example: Write down the place value of each digit in the number 543.

Solution: The correct place value of each digit in the number can be expressed as follows:

• 5 × 100 = 500 or 5 hundreds
• 4 × 10 = 40 or 4 tens
• 3 × 1 = 3 or 3 ones
• 

1.Relative error

The relative error is defined as the ratio of the absolute error of the measurement to the actual measurement. Using this method, we can determine the magnitude of the absolute error in terms of the actual size of the measurement.

How do you find the relative error? The relative error is found by dividing the absolute error by the measured value. The relative error equation is: Relative error = absolute error / measured value.

Eg: A student measures the width of a piece of paper to be 36 centimeters. The actual width of the paper is 35.75 centimeters. What is the relative error in this measurement?

Ans: In this case, it is |36 – 35.75| = 0.25 cm. Next, we calculate the relative error. The relative error is the absolute error divided by the actual value, then multiplied by 100% to convert it to a percentage. So, the relative error is (0.25 / 35.75) * 100% = 0.7%.

The Mathematics Test for teachers is one of the Professional teacher’s license requirements for those who are teaching from grades 9 through 12 in mathematics.

Test Name Mathematics Test Grade 9-12

Number of questions 100

Test Duration 2 and ½ hours

 Format of questions Multiple Choice questions/ Fill in the blank

 Test Delivery Computer delivered

The Mathematics Test for teachers is one of the Professional teacher’s license requirements for those who are teaching from grades 3 through 8 in mathematics.

Test Overview

Test Name Mathematics Test Grade 3-8

Number of questions 100

Test Duration 2 and ½ hours

 Format of questions Multiple Choice questions/ Fill in the blank

 Test Delivery Computer delivered