The ratio of balance in the savings accounts of Arielle, Barry and Chanelle is $9:5:4$ . After Arielle transfer $\$180$ to Chanelle, the ratio becomes $27:20:25$ . Find the original balance in the savings account of Ariell, Barry and Chanelle

View step-by-step help

Home

Math Problems

Grade 7

Write an equivalent ratio

Full solution

Q. The ratio of balance in the savings accounts of Arielle, Barry and Chanelle is

9:5:4

. After Arielle transfer

\$180

to Chanelle, the ratio becomes

27:20:25

. Find the original balance in the savings account of Ariell, Barry and Chanelle

Original Balance Ratios: Let's call Arielle's original balance $A$ , Barry's $B$ , and Chanelle's $C$ . The original ratio is $A:B:C = 9:5:4$ .
New Ratio Calculation: We know that after Arielle transfers $\$180$ to Chanelle, the new ratio is $27:20:25$ . This means Arielle's new balance is $A - 180$ , and Chanelle's new balance is $C + 180$ .
Proportion Setup: The new ratio can be written as $A - 180$ : $B$ : $C + 180$ = $27$ : $20$ : $25$ . Since the ratios are equivalent, we can set up a proportion: $\frac{A - 180}{27}$ = $\frac{B}{20}$ = $\frac{C + 180}{25}$ .
Common Multiplier Calculation: Let's find a common multiplier for the denominators $27$ , $20$ , and $25$ . The least common multiple (LCM) of $27$ , $20$ , and $25$ is $5400$ . So, we can multiply each part of the proportion by $5400$ to get rid of the denominators.
Equation Simplification: Multiplying each part by $5400$ gives us: $5400\times(A - 180)/27 = 5400\times B/20 = 5400\times(C + 180)/25$ . Simplifying, we get $200\times(A - 180) = 270\times B = 216\times(C + 180)$ .
Original Ratio Equations: Now, let's look at the original ratio $9:5:4$ . We can say that $\frac{A}{9} = \frac{B}{5} = \frac{C}{4}$ . Using the LCM of $9$ , $5$ , and $4$ , which is $180$ , we multiply each part by $180$ to get: $20A = 36B = 45C$ .
Expressing B and C in Terms of A: We have two sets of equations now: $200*(A - 180) = 270*B = 216*(C + 180)$ and $20*A = 36*B = 45*C$ . We can use these to find the values of $A$ , $B$ , and $C$ .
Substitution into Equations: From the second set of equations, we can express $B$ and $C$ in terms of $A$ : $B = \frac{20}{36}A$ and $C = \frac{20}{45}A$ .
Equation Simplification Correction: Substitute $B$ and $C$ into the first set of equations: $200\times(A - 180) = 270\times\left(\frac{20}{36}\right)A = 216\times\left(\frac{20}{45}\right)A + 180$ .
Equation Simplification Correction: Substitute $B$ and $C$ into the first set of equations: $200*(A - 180) = 270*((20/36)*A) = 216*((20/45)*A + 180)$ . Simplify the equations: $200*A - 36000 = 150*A = 96*A + 38880$ . Oops, there's a mistake here. The equation should maintain the balance between all three parts, but it doesn't. We need to correct this.