Math Problems: Addition, Subtraction, Multiplication, Division
Hey guys! Today, we're diving into some cool math problems involving addition, subtraction, multiplication, and division. We'll tackle both regular calculations and those using scientific notation. So, grab your calculators and let's get started!
1. Basic Arithmetic Operations
Let's kick things off with some fundamental arithmetic. It’s super important to master these because they form the bedrock for more complex calculations. We'll break down each part step by step, making sure everyone's on the same page.
a) 112.6 m + 8,005 m + 13.48 m
When it comes to addition, the key is to line up those decimal points! This ensures we're adding the correct place values together. So, let’s align these numbers:
112.60 m
8005.00 m
+ 13.48 m
----------
Now, let's add 'em up:
112.60 m
8005.00 m
+ 13.48 m
----------
8131.08 m
So, 112.6 m + 8,005 m + 13.48 m equals 8131.08 m. See? Not so scary when we take it one step at a time!
b) 78.05 cm² - 32.046 cm²
Subtraction is the name of the game here! Just like with addition, aligning those decimal points is crucial. This helps us keep track of what we're subtracting from where. Let's set it up:
78.050 cm²
- 32.046 cm²
----------
Notice we added a zero to 78.05 to make the subtraction easier. Now, let’s subtract:
78.050 cm²
- 32.046 cm²
----------
46.004 cm²
Therefore, 78.05 cm² - 32.046 cm² gives us 46.004 cm². We're on a roll!
c) 0.1682 m x 8.2 m
Time for multiplication! When multiplying decimals, we don’t worry about aligning the decimal points initially. We just multiply as if they were whole numbers, and then we count the total number of decimal places in the original numbers to determine where to place the decimal point in our answer.
Let’s multiply 1682 by 82:
1682
x 82
------
3364
134560
------
137924
Now, let's count the decimal places. 0.1682 has four decimal places, and 8.2 has one, giving us a total of five decimal places. So, we place the decimal point five places from the right in our result:
- 1682 m x 8.2 m = 1.37924 m². Multiplication, conquered!
d) 94.5 / 1.2 s
Alright, let's tackle some division! Dividing decimals can seem tricky, but we can make it simpler by getting rid of the decimal in the divisor. We do this by multiplying both the dividend and the divisor by a power of 10 that will turn the divisor into a whole number.
In this case, we multiply both 94.5 and 1.2 by 10:
94. 5 / 1.2 = (94.5 x 10) / (1.2 x 10) = 945 / 12
Now, let's divide 945 by 12:
78.75
12 | 945.00
- 84
----
105
- 96
----
90
- 84
----
60
- 60
----
0
So, 94.5 / 1.2 s equals 78.75 s. Division? Nailed it!
e) 7,500 x 10³ kg
This one’s a bit different! We've got scientific notation sneaking in here. But don’t worry, it's easier than it looks. 10³ simply means 10 to the power of 3, which is 1,000. So, we're really just multiplying 7,500 by 1,000:
7,500 x 10³ kg = 7,500 x 1,000 kg = 7,500,000 kg
Easy peasy, right?
f) 510 x 10³ m x 510 m x 4 m
Okay, let's break this down. First, let’s deal with that scientific notation again. 510 x 10³ m means 510 multiplied by 1,000, which is 510,000 m. Now we have:
510,000 m x 510 m x 4 m
Let's multiply these numbers:
510,000 m x 510 m = 260,100,000 m²
Now multiply by 4 m:
260,100,000 m² x 4 m = 1,040,400,000 m³
Wow, that’s a big number! But we handled it like pros.
2. Scientific Notation Calculations
Now, let’s level up and work with some calculations entirely in scientific notation. Scientific notation is super handy for dealing with very large or very small numbers. It makes them much easier to handle and understand. Remember, the general form is a x 10^b, where 'a' is a number between 1 and 10, and 'b' is an integer.
a) 2.46 x 10³ g + 5.4 x 10³ g
When adding numbers in scientific notation, the exponents must be the same. Lucky for us, they already are! So, we can simply add the numbers in front (the coefficients):
(2.46 + 5.4) x 10³ g = 7.86 x 10³ g
So, 2.46 x 10³ g + 5.4 x 10³ g equals 7.86 x 10³ g. That was a smooth one!
b) 5.80 x 10⁹ s + 3.20 x 10⁸ s
Here, the exponents are different, so we need to make them the same before adding. We can convert 3.20 x 10⁸ s to have an exponent of 9 by moving the decimal point one place to the left:
- 20 x 10⁸ s = 0.320 x 10⁹ s
Now we can add:
(5.80 x 10⁹ s) + (0.320 x 10⁹ s) = (5.80 + 0.320) x 10⁹ s = 6.12 x 10⁹ s
Therefore, 5.80 x 10⁹ s + 3.20 x 10⁸ s equals 6.12 x 10⁹ s. A little exponent juggling, but we got there!
c) 5.87 x 10⁻⁴ m - 2.83 x 10⁻⁶ m
Time for subtraction with scientific notation! Again, we need the exponents to match. Let’s convert 2.83 x 10⁻⁶ m to have an exponent of -4. To do this, we move the decimal point two places to the left:
- 83 x 10⁻⁶ m = 0.0283 x 10⁻⁴ m
Now we subtract:
(5.87 x 10⁻⁴ m) - (0.0283 x 10⁻⁴ m) = (5.87 - 0.0283) x 10⁻⁴ m = 5.8417 x 10⁻⁴ m
So, 5.87 x 10⁻⁴ m - 2.83 x 10⁻⁶ m is 5.8417 x 10⁻⁴ m. Subtraction? Sorted!
d) (5.60 x 10⁷ g) / (2.8 x 10⁻² cm³)
Last but not least, let's do some division with scientific notation. When dividing, we divide the coefficients and subtract the exponents:
(5.60 / 2.8) x 10^(7 - (-2)) g/cm³ = 2.0 x 10^(7 + 2) g/cm³ = 2.0 x 10⁹ g/cm³
Thus, (5.60 x 10⁷ g) / (2.8 x 10⁻² cm³) equals 2.0 x 10⁹ g/cm³. Division? Done and dusted!
Wrapping Up
Phew! We've covered a lot today, from basic arithmetic to scientific notation. Remember, the key to mastering math is practice, practice, practice! So keep crunching those numbers, and you'll be a math whiz in no time. Keep your head up, you can do this!