Solve 65%*30+25%*18+(1-65%-25%)(a-90)/90geq25%

$$\frac { 65 \% \times 30 + 25 \% \times 18 + ( 1 - 65 \% - 25 \% ) ( a - 90 ) } { 90 } \geq 25 \%$$

Solve for a

$$a\geq 75$$

Solution Steps

Multiply both sides of the equation by 900, the least common multiple of 90,100. Since 900 is positive, the inequality direction remains the same.

$$10\left(\frac{65}{100}\times 30+\frac{25}{100}\times 18+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{65}{100} to lowest terms by extracting and canceling out 5.

$$10\left(\frac{13}{20}\times 30+\frac{25}{100}\times 18+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Express \frac{13}{20}\times 30 as a single fraction.

$$10\left(\frac{13\times 30}{20}+\frac{25}{100}\times 18+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Multiply 13 and 30 to get 390.

$$10\left(\frac{390}{20}+\frac{25}{100}\times 18+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{390}{20} to lowest terms by extracting and canceling out 10.

$$10\left(\frac{39}{2}+\frac{25}{100}\times 18+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{25}{100} to lowest terms by extracting and canceling out 25.

$$10\left(\frac{39}{2}+\frac{1}{4}\times 18+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Multiply \frac{1}{4} and 18 to get \frac{18}{4}.

$$10\left(\frac{39}{2}+\frac{18}{4}+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$10\left(\frac{39}{2}+\frac{9}{2}+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Since \frac{39}{2} and \frac{9}{2} have the same denominator, add them by adding their numerators.

$$10\left(\frac{39+9}{2}+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Add 39 and 9 to get 48.

$$10\left(\frac{48}{2}+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Divide 48 by 2 to get 24.

$$10\left(24+\left(1-\frac{65}{100}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{65}{100} to lowest terms by extracting and canceling out 5.

$$10\left(24+\left(1-\frac{13}{20}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Convert 1 to fraction \frac{20}{20}.

$$10\left(24+\left(\frac{20}{20}-\frac{13}{20}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Since \frac{20}{20} and \frac{13}{20} have the same denominator, subtract them by subtracting their numerators.

$$10\left(24+\left(\frac{20-13}{20}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Subtract 13 from 20 to get 7.

$$10\left(24+\left(\frac{7}{20}-\frac{25}{100}\right)\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{25}{100} to lowest terms by extracting and canceling out 25.

$$10\left(24+\left(\frac{7}{20}-\frac{1}{4}\right)\left(a-90\right)\right)\geq 9\times 25$$

Least common multiple of 20 and 4 is 20. Convert \frac{7}{20} and \frac{1}{4} to fractions with denominator 20.

$$10\left(24+\left(\frac{7}{20}-\frac{5}{20}\right)\left(a-90\right)\right)\geq 9\times 25$$

Since \frac{7}{20} and \frac{5}{20} have the same denominator, subtract them by subtracting their numerators.

$$10\left(24+\frac{7-5}{20}\left(a-90\right)\right)\geq 9\times 25$$

Subtract 5 from 7 to get 2.

$$10\left(24+\frac{2}{20}\left(a-90\right)\right)\geq 9\times 25$$

Reduce the fraction \frac{2}{20} to lowest terms by extracting and canceling out 2.

$$10\left(24+\frac{1}{10}\left(a-90\right)\right)\geq 9\times 25$$

Use the distributive property to multiply \frac{1}{10} by a-90.

$$10\left(24+\frac{1}{10}a+\frac{1}{10}\left(-90\right)\right)\geq 9\times 25$$

Multiply \frac{1}{10} and -90 to get \frac{-90}{10}.

$$10\left(24+\frac{1}{10}a+\frac{-90}{10}\right)\geq 9\times 25$$

Divide -90 by 10 to get -9.

$$10\left(24+\frac{1}{10}a-9\right)\geq 9\times 25$$

Subtract 9 from 24 to get 15.

$$10\left(15+\frac{1}{10}a\right)\geq 9\times 25$$

Use the distributive property to multiply 10 by 15+\frac{1}{10}a.

$$150+10\times \frac{1}{10}a\geq 9\times 25$$

Cancel out 10 and 10.

$$150+a\geq 9\times 25$$

Multiply 9 and 25 to get 225.

$$150+a\geq 225$$

Subtract 150 from both sides.

$$a\geq 225-150$$

Subtract 150 from 225 to get 75.

$$a\geq 75$$