Maths Olympiad Prep

Track / Stage 3 / 161 of 260 #161 of 1964

Problem 161

AMC 10/12, early questions
Algebra Difficulty 3.4 Find the answer

Calculate the following:(1)2.4+3.54.6+3.5(1)-2.4+3.5-4.6+3.5;(2)(40)(28)(19)+(24)(2)\left(-40\right)-\left(-28\right)-\left(-19\right)+\left(-24\right);(3)(3)×56×(45)×(14)(3)(-3)×\frac{5}{6}×(-\frac{4}{5})×(-\frac{1}{4});(4)(57)×(43)÷(217)(4)(-\frac{5}{7})×(-\frac{4}{3})÷(-2\frac{1}{7}).

A number or a short expression. Spacing and $ signs are ignored.

Official solution

To solve each of the given problems step by step, we follow the given solutions closely, breaking down the calculations in a detailed manner.

### Problem 1:
Calculate 2.4+3.54.6+3.5-2.4 + 3.5 - 4.6 + 3.5.

We start by grouping the positive and negative numbers:
2.4+3.54.6+3.5=(2.44.6)+(3.5+3.5)=7+7=0. \begin{align*} -2.4 + 3.5 - 4.6 + 3.5 &= (-2.4 - 4.6) + (3.5 + 3.5) \\ &= -7 + 7 \\ &= 0. \end{align*}
Thus, the answer is 0\boxed{0}.

### Problem 2:
Calculate (40)(28)(19)+(24)(-40) - (-28) - (-19) + (-24).

We simplify the expression by changing the double negatives to positives:
(40)(28)(19)+(24)=(40)+28+19+(24)=17. \begin{align*} (-40) - (-28) - (-19) + (-24) &= (-40) + 28 + 19 + (-24) \\ &= -17. \end{align*}
So, the answer is 17\boxed{-17}.

### Problem 3:
Calculate (3)×56×(45)×(14)(-3) \times \frac{5}{6} \times (-\frac{4}{5}) \times (-\frac{1}{4}).

We multiply the fractions step by step:
3×56×45×14=12. \begin{align*} -3 \times \frac{5}{6} \times \frac{4}{5} \times \frac{1}{4} &= -\frac{1}{2}. \end{align*}
Therefore, the answer is 12\boxed{-\frac{1}{2}}.

### Problem 4:
Calculate (57)×(43)÷(217)(-\frac{5}{7}) \times (-\frac{4}{3}) ÷ (-2\frac{1}{7}).

First, we convert the mixed number to an improper fraction:
217=157. -2\frac{1}{7} = -\frac{15}{7}.

Then, we perform the multiplication and division:
57×43×715=49. \begin{align*} -\frac{5}{7} \times \frac{4}{3} \times \frac{7}{15} &= -\frac{4}{9}. \end{align*}
Hence, the final answer is 49\boxed{-\frac{4}{9}}.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.