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Simplify the following expressions: 600 + 52 + 25 * 3 - 45 + 22, Find the cost of 130 apples if 110 apples cost Rs. 1320 and the price increased by 10%. Find the lengths of the sides of a rectangle wi

Mathematics

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  1. Evaluate the exponents.

    600+52+25×3−45+22=600+25+25×3−45+4600 + 5^2 + 25 \times 3 - 45 + 2^2 = 600 + 25 + 25 \times 3 - 45 + 4

  2. Perform the multiplication.

    600+25+25×3−45+4=600+25+75−45+4600 + 25 + 25 \times 3 - 45 + 4 = 600 + 25 + 75 - 45 + 4

  3. Perform addition and subtraction from left to right.

    600+25+75−45+4=625+75−45+4600 + 25 + 75 - 45 + 4 = 625 + 75 - 45 + 4 625+75−45+4=700−45+4625 + 75 - 45 + 4 = 700 - 45 + 4 700−45+4=655+4700 - 45 + 4 = 655 + 4 655+4=659655 + 4 = 659

    The simplified value is 659\boxed{\text{659}}.

    2.

  4. Calculate the original price per apple.

    Price per apple=Total costNumber of apples=Rs. 1320110=Rs. 12 per apple\text{Price per apple} = \frac{\text{Total cost}}{\text{Number of apples}} = \frac{\text{Rs. } 1320}{110} = \text{Rs. } 12 \text{ per apple}

  5. Calculate the new price per apple after a 10% increase.

    Increase amount=10%×Rs. 12=0.10×Rs. 12=Rs. 1.20\text{Increase amount} = 10\% \times \text{Rs. } 12 = 0.10 \times \text{Rs. } 12 = \text{Rs. } 1.20 New price per apple=Rs. 12+Rs. 1.20=Rs. 13.20\text{New price per apple} = \text{Rs. } 12 + \text{Rs. } 1.20 = \text{Rs. } 13.20

  6. Calculate the cost of 130 apples at the new price.

    Cost of 130 apples=130×Rs. 13.20=Rs. 1716\text{Cost of 130 apples} = 130 \times \text{Rs. } 13.20 = \text{Rs. } 1716

    The cost of 130 apples will be Rs. 1716\boxed{\text{Rs. 1716}}.

    3.

  7. Determine the lengths of the sides of the rectangle.

    The vertices are A(12,3), B(12,7), C(15,7), D(15,3).
    Length AB (vertical side): The x-coordinate is constant (12). The length is the difference in y-coordinates: 7−3=47 - 3 = 4 units.
    Length BC (horizontal side): The y-coordinate is constant (7). The length is the difference in x-coordinates: 15−12=315 - 12 = 3 units.
    So, the length of the rectangle is 4 units and the width is 3 units.

  8. Calculate the perimeter in units.

    Perimeter in units=2×(Length+Width)=2×(4+3)=2×7=14 units\text{Perimeter in units} = 2 \times (\text{Length} + \text{Width}) = 2 \times (4 + 3) = 2 \times 7 = 14 \text{ units}

  9. Convert the perimeter from units to cm.

    Given that 1 unit = 2 cm.

    Perimeter in cm=14 units×2 cm/unit=28 cm\text{Perimeter in cm} = 14 \text{ units} \times 2 \text{ cm/unit} = 28 \text{ cm}

    The perimeter of the rectangle is 28 cm\boxed{\text{28 cm}}.

    4.

  10. Set up a proportion for the ratio of orange juice to pineapple juice.

    Orange juicePineapple juice=52\frac{\text{Orange juice}}{\text{Pineapple juice}} = \frac{5}{2}

  11. Substitute the given amount of orange juice and solve for pineapple juice.

    450 mlP=52\frac{450 \text{ ml}}{P} = \frac{5}{2} 5P=450 ml×25P = 450 \text{ ml} \times 2 5P=900 ml5P = 900 \text{ ml} P=900 ml5P = \frac{900 \text{ ml}}{5} P=180 mlP = 180 \text{ ml}

    You need 180 ml\boxed{\text{180 ml}} of pineapple juice.

    5.

  12. Calculate the total shift time in minutes.

    Total shift time=10 hours×60 minutes/hour=600 minutes\text{Total shift time} = 10 \text{ hours} \times 60 \text{ minutes/hour} = 600 \text{ minutes}

  13. Calculate the time spent on each activity in minutes.

    • Checking emails: 130×600 minutes=20 minutes\frac{1}{30} \times 600 \text{ minutes} = 20 \text{ minutes}
    • Data entry: 24%×600 minutes=0.24×600 minutes=144 minutes24\% \times 600 \text{ minutes} = 0.24 \times 600 \text{ minutes} = 144 \text{ minutes}
    • Web designing: 12×600 minutes=300 minutes\frac{1}{2} \times 600 \text{ minutes} = 300 \text{ minutes}
    • Project discussion: 4.06%×600 minutes=0.0406×600 minutes=24.36 minutes4.06\% \times 600 \text{ minutes} = 0.0406 \times 600 \text{ minutes} = 24.36 \text{ minutes}

  14. Calculate the total time spent on these activities.

    Total activity time=20+144+300+24.36=488.36 minutes\text{Total activity time} = 20 + 144 + 300 + 24.36 = 488.36 \text{ minutes}

  15. Calculate the remaining time for rest and lunch break.

    Remaining time=Total shift time−Total activity time\text{Remaining time} = \text{Total shift time} - \text{Total activity time} Remaining time=600 minutes−488.36 minutes=111.64 minutes\text{Remaining time} = 600 \text{ minutes} - 488.36 \text{ minutes} = 111.64 \text{ minutes}

    Harish spends 111.64 minutes\boxed{\text{111.64 minutes}} on rest and lunch break.

    6.

  16. Identify the relationship between the angles.

    The angle 125∘125^\circ is an exterior angle of the triangle. The interior angle adjacent to it is 180∘−125∘=55∘180^\circ - 125^\circ = 55^\circ.

  17. Apply the angle sum property of a triangle.

    The sum of the interior angles of a triangle is 180∘180^\circ.

    x+80∘+55∘=180∘x + 80^\circ + 55^\circ = 180^\circ x+135∘=180∘x + 135^\circ = 180^\circ x=180∘−135∘x = 180^\circ - 135^\circ x=45∘x = 45^\circ

    Alternatively, using the exterior angle theorem: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

    125∘=x+80∘125^\circ = x + 80^\circ x=125∘−80∘x = 125^\circ - 80^\circ x=45∘x = 45^\circ

    The value of xx is 45∘\boxed{45^\circ}.

    7.

  18. Set up a proportion for the ratio of flour to sugar.

    FlourSugar=45\frac{\text{Flour}}{\text{Sugar}} = \frac{4}{5}

  19. Substitute the given amount of sugar and solve for flour.

    F300 g=45\frac{F}{300 \text{ g}} = \frac{4}{5} 5F=4×300 g5F = 4 \times 300 \text{ g} 5F=1200 g5F = 1200 \text{ g} F=1200 g5F = \frac{1200 \text{ g}}{5} F=240 gF = 240 \text{ g}

    Jay used 240 g\boxed{\text{240 g}} of flour.

    8.

  20. Calculate the area of the circular garden.

    Given radius r=7r = 7 cm and π=3.14\pi = 3.14.

    Area of circle=πr2=3.14×(7 cm)2=3.14×49 cm2=153.86 cm2\text{Area of circle} = \pi r^2 = 3.14 \times (7 \text{ cm})^2 = 3.14 \times 49 \text{ cm}^2 = 153.86 \text{ cm}^2

  21. Calculate the area of the rectangular garden.

    Given length l=20l = 20 cm and width w=8w = 8 cm.

    Area of rectangle=l×w=20 cm×8 cm=160 cm2\text{Area of rectangle} = l \times w = 20 \text{ cm} \times 8 \text{ cm} = 160 \text{ cm}^2

  22. Compare the areas.

    Circular garden area = 153.86 cm2153.86 \text{ cm}^2
    Rectangular garden area = 160 cm2160 \text{ cm}^2
    Since 160 cm2>153.86 cm2160 \text{ cm}^2 > 153.86 \text{ cm}^2, the rectangular garden has more area.
    Option b) A rectangular garden with dimensions 20 cm×8 cm\boxed{\text{b) A rectangular garden with dimensions 20 cm} \times \text{8 cm}} gives more area.

    9.
    To find the coldest temperature, we need to find the smallest number in the given list: -3°C, -23°C, 56°C, -47°C, -45°C, 0°C.
    Comparing the numbers:
    Positive temperatures: 56°C, 0°C
    Negative temperatures: -3°C, -23°C, -47°C, -45°C
    The smallest number among these is the one furthest from zero in the negative direction.
    -47 is smaller than -45, -23, and -3.
    The coldest temperature is -47°C\boxed{\text{-47°C}}.

    10.

  23. Calculate the time taken using the formula Time=DistanceSpeedTime = \frac{Distance}{Speed}.

    Time=97.5 miles26 mph=3.75 hours\text{Time} = \frac{97.5 \text{ miles}}{26 \text{ mph}} = 3.75 \text{ hours}

  24. Convert the decimal part of the hours into minutes.

    0.75 hours×60 minutes/hour=45 minutes0.75 \text{ hours} \times 60 \text{ minutes/hour} = 45 \text{ minutes}

    So, the total time is 3 hours and 45 minutes.
    The journey will be completed in 3 hours and 45 minutes\boxed{\text{3 hours and 45 minutes}}.

    11.

  25. Substitute the given values of xx and yy into the expression.

    Given x=35x = 35 and y=3.5y = 3.5.

    24x+350y=24(35)+350(3.5)24x + 350y = 24(35) + 350(3.5)

  26. Perform the multiplications.

    24×35=84024 \times 35 = 840 350×3.5=1225350 \times 3.5 = 1225

  27. Perform the addition.

    840+1225=2065840 + 1225 = 2065

    The value of 24x+350y24x + 350y is 2065\boxed{2065}.

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