If the sum of two numbers is 10 and the sum of their reciprocals is 5/12 the numbers would be

5 Answers

Let A , and B represent the numbers. A + B = 10 1/A + 1/B = 5/12 Find the common denominator for A, B which is AB by multiplying B by 1 and A, and multiplying A by 1 and B The second equation will be (A+B)/AB = 5/12 we know that A+B = 10 You now have 10/AB = 5/12 cross multiply and solve for AB 10x12=5AB120 = 5AB divide both sides by 5 24 = AB We now have A + B = 10 AB = 24Divide both sides by B A = 24/BSubstitute A in Equation 1 24/B + B =10Multiply each term by B24 + B^2 = 10BSubtract 10B from both sidesB^2 -10B +24 = 0(B-6)(B-4) = 0 B= 6, and B=4 If B = 6, then A =4 Your two numbers are 6 and 4.

(B + A)/AB = 5/12

Let the two numbers are a and b.Then, a+b = 10......(1)1/a+1/b = 5/12(a+b)/ab = 5/1212(a+b) = 5ab....(2)substitute the value from eq. (1) into eq. (2)12(10) = 5ab5ab = 120ab = 24a = 24/b....(3)substitute the value of a from eq. (3) into eq. (1)(24/b)+b=1024+b^2= 10bb^2-10b+24=0b^2-6b-4b+24=0b(b-6)-4(b-6)=0(b-4)(b-6)=0b = 4,6Put the value of b into eq. (1)

a = 6,4

Let two numbers are a and bthen, a+b=101/a+1/b=5/12Solving, (a+b)/ab=5/1212(a+b)=5abAfter putting the value of a+b12(10)=5ab5ab=120ab=24a=24/bAfter putting the velue of a(24/b)+b=10b^2-10b+24=0b^2-6b-4b+24=0b(b-6)-4(b-6)(b-4)(b-6)b=4,6a=6,4

so, numbers are (6,4) and (4,6)

Let two numbers are x and ythen, x+y= 101/x+1/y=5/12Then Second equation will be,(x+y)/xy=5/1210/xy=5/125xy=120xy=24x=24/ythen, 24/y+y=10y^2-10y+24=0y^2-6y-4y+24=0y(y-6)-4(y-6)=0(y-4)(y-6)=0y=4,6then, x=12,4

So numbers are (4,6)

Let two numbers are a and bthen, a+b=10....(1)1/a+1/b=5/12then, (a+b)/ab = 5/12....(2)putting the value from eq. (1) in eq. (2)10/ab = 5/125ab = 120ab = 24a=24/b....(3)putting the value of a from eq. (3) to (1)(24/b)+b=10b^2-10b+24=0b^2-6b-4b+24=0b(b-6)-4(b-6)=0(b-4)(b-6)=0b=4,6then, a=6,4

Then, pairs are (6,4) and (4,6)

Free

10 Qs. 10 Marks 10 Mins

Given:

Sum of two numbers = 10

Sum of their reciprocals = 5/12

Calculation:

Let the numbers be x and y

So, x + y = 10 ----(i)

And, (1/x) + (1/y) = 5/12 ----(ii)

⇒ (x + y)/xy = 5/12

Now,

⇒ 10/xy = 5/12

⇒ xy = 12 × 2 = 24 ----(iii)

Now, (10 – y) × y = 24

⇒ 10y – y2 = 24

⇒ y2 – 10y + 24 = 0

⇒ y2 – 6y – 4y + 24 = 0

⇒ y(y – 6) – 4(y – 6) = 0

⇒ (y – 6) (y – 4) = 0

⇒ y = 6, 4

∴ The required numbers are 6 and 4

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The two numbers are 4 and 6.

Of course you could guess and check which two numbers make 10 (e.g. 1&9, 2&8,...5&5) but the statement isn't clear if the number is positive, whole, fraction, rational, etc. which could take forever so let's derive using math and not guess.

So the two statements give two equations, which should be enough to figure out the number. First equation is X+Y=10. Second equation is (1/X)+(1/Y)=(5/12). Taking the first equation and rearranging to one variable, we get X=10-Y. Substituting the rearranged first equation into the second equation, we get (1/(10-Y))+(1/Y)=(5/12).

The revised second equation does not have a common denominator so we'll need to multiple by a 'theoretical 1' to get a common denominator. The first term will need to be multiplied by (Y/Y) and the second term will be multiplied by ((10-Y)/(10/Y)). The right hand side (5/12) remains the same since we technically multiplied the left side individual terms by a theoretical 1. The left side is now (Y+(10-Y))/(-Y2+10Y). Simplifying the numerator, we get 10. The equation now reads 10/(-Y2+10Y)=(5/12). Cross multiplying, we get 120 = -5Y2 + 50Y. This is a quadratic equation (i.e. single variable squared). Rearrange to get all terms on one side and zero on the other side. Thus 5Y2 - 50Y - 120 = 0. Use the quadratic equation (a=5, b=-50, c=120) and solve for Y. Thus, Y is equal to 4 and 6.

Going back to the original equation, plug in Y to the easier equation. Let's assume Y=4. Then we quickly see X=6 (based on the first simple equation). If we assumed Y=4, then X=4. Essentially no difference.

We need to check the second equation to truly verify our answer. (1/4)+(1/6) need to have common denominators so 12 is the best LCM (Lowest Common Denominator). Multiply each term by their respective 'theoretical 1' so (3/3)*(1/4) + (2/2)*(1/6) is equal to (3+2)/12, which is 5/12.