1.)A picture 8 in. by 12 in. is placed in a frame which has a uniform width. if the area of the frame is equal to the area of the picture, find the dimension of the frame
2.)Arman has 1160 annual income from bonds bearing 3% and 5 % interest. then he added 25% more of the 3% bonds and 40% more of the 5 % bonds, thereby increasing his annual income by 410. find his initial investment in each type of bond
i have a couple of problems like these,please help--show me the formulas and solutions to these so i can study how it's done
thank YOU so much
1.)
width= 8+2x
length= 12 + 2x
(8+2x)(12+2x)-96=96
now simplify
Sure, I'd be happy to help you with these problems! Let's tackle them one by one:
1.) To find the dimensions of the frame, we need to set up an equation based on the given information. Let's assume the width of the frame is 'x'.
The area of the picture is 8 in. by 12 in., which means its area is 8 * 12 = 96 square inches.
The area of the frame can be calculated by subtracting the area of the picture from the area of the frame plus picture. The total area of the frame plus picture will be (8 + 2x) * (12 + 2x), where 2x represents the width of the frame on each side. So, the area of the frame is:
(8 + 2x) * (12 + 2x) - 96
Since we're given that the area of the frame is equal to the area of the picture, we can set up the equation:
(8 + 2x) * (12 + 2x) - 96 = 96
Simplifying this equation will give us the value of 'x', which is the width of the frame. Once we find 'x', we can calculate the dimensions of the frame by adding 'x' to both the length and width of the picture.
2.) To find Arman's initial investment in each type of bond, we can set up an equation based on the given information.
Let's assume Arman initially invested 'x' amount of money in the 3% bonds. So his investment in the 5% bonds would be (1160 - x) (since his total annual income is 1160).
When he adds 25% more of the 3% bonds, his new investment in the 3% bonds becomes 1.25x. Similarly, when he adds 40% more of the 5% bonds, his new investment in the 5% bonds becomes 1.4(1160 - x).
Adding these two new investments together should give us his new total annual income, which is 1160 + 410. So we can set up the equation:
1.25x + 1.4(1160 - x) = 1160 + 410
Simplifying this equation will give us the value of 'x', which is Arman's initial investment in the 3% bonds. Once we find 'x', we can calculate his initial investment in the 5% bonds by subtracting 'x' from 1160.
To solve these equations, you can use algebraic techniques such as expanding brackets, simplifying and rearranging terms, and solving linear equations.
I hope this helps! Let me know if there's anything else I can assist you with.